EXHIBIT 4.9

          __________________________________________________________


                              GUARANTEE AGREEMENT


                                    BETWEEN


                              GREATER BAY BANCORP
                                (AS GUARANTOR)


                                      AND


                           WILMINGTON TRUST COMPANY
                                 (AS TRUSTEE)



                                  DATED AS OF
                                        
                                MARCH 31, 1997



          __________________________________________________________

 
                             CROSS-REFERENCE TABLE*



         Section of Trust                                      Section of
Indenture Act of 1939, as amended                          Guarantee Agreement
- ---------------------------------                          -------------------
                                                          
              310(a).                                              4.1(a)     
              310(b).                                            4.1(c), 2.8  
              310(c).                                           Inapplicable  
              311(a).                                              2.2(b)     
              311(b).                                              2.2(b)     
              311(c).                                           Inapplicable  
              312(a).                                              2.2(a)     
              312(b).                                              2.2(b)     
               313.                                                  2.3      
              314(a).                                                2.4      
              314(b).                                           Inapplicable  
              314(c).                                                2.5      
              314(d).                                           Inapplicable  
              314(e).                                           1.1, 2.5, 3.2 
              314(f).                                             2.1, 3.2    
              315(a).                                              3.1(d)     
              315(b).                                                2.7      
              315(c).                                                3.1      
              315(d).                                              3.1(d)     
              316(a).                                           1.1, 2.6, 5.4 
              316(b).                                                5.3      
              316(c).                                                9.2      
              317(a).                                           Inapplicable  
              317(b).                                           Inapplicable  
              318(a).                                              2.1(b)     
              318(b).                                                2.1      
              318(c).                                              2.1(a)      


- ----------------
*  This Cross-Reference Table does not constitute part of the Guarantee
   Agreement and shall not affect the interpretation of any of its terms or
   provisions.

 
                               TABLE OF CONTENTS


                                                                          Page
                                                                          ----
                                                                       
ARTICLE I.   DEFINITIONS                                                     1

          Section 1.1. Definitions                                           1
 

ARTICLE II.  TRUST INDENTURE ACT                                             3

          Section 2.1. Trust Indenture Act; Application                      3

          Section 2.2. List of Holders                                       4

          Section 2.3. Reports by the Guarantee Trustee                      4

          Section 2.4. Periodic Reports to Guarantee Trustee                 4

          Section 2.5. Evidence of Compliance with Conditions Precedent      4

          Section 2.6. Events of Default; Waiver                             4

          Section 2.7. Event of Default; Notice                              4

          Section 2.8. Conflicting Interests                                 5
 

ARTICLE III. POWERS, DUTIES AND RIGHTS OF THE GUARANTEE                      5
             TRUSTEE

          Section 3.1. Powers and Duties of the Guarantee Trustee            5

          Section 3.2. Certain Rights of Guarantee Trustee                   6

          Section 3.3. Indemnity                                             8

 
ARTICLE IV.  GUARANTEE TRUSTEE                                               8

          Section 4.1. Guarantee Trustee; Eligibility                        8

          Section 4.2. Appointment, Removal and Resignation of the 
                       Guarantee Trustee                                     8
 

 
 
                                                                       
ARTICLE V.   GUARANTEE                                                       8

          Section 5.1. Guarantee                                             8

          Section 5.2. Waiver of Notice and Demand                           9

          Section 5.3. Obligations Not Affected                              9

          Section 5.4. Rights of Holders                                    10

          Section 5.5. Guarantee of Payment                                 10

          Section 5.6. Subrogation                                          10

          Section 5.7. Independent Obligations                              10
 

ARTICLE VI.  COVENANTS AND SUBORDINATION                                    11

          Section 6.1. Subordination                                        11

          Section 6.2. Pari Passu Guarantees                                11
 

ARTICLE VII.  CONSOLIDATION, MERGER, SALE OF ASSETS AND OTHER 
              TRANSACTIONS                                                  11

          Section 7.1. Guarantor May Consolidate, Etc., Only on 
                       Certain Terms                                        11

          Section 7.2. Successor Guarantor Substituted                      12
 

ARTICLE VIII. TERMINATION                                                   12

          Section 8.1. Termination                                          12
                                                                            
 
ARTICLE IX.   MISCELLANEOUS                                                 12

          Section 9.1. Successors and Assigns                               12

          Section 9.2. Amendments                                           12

          Section 9.3. Notices                                              12
 

 
 
                                                                       
          Section 9.4. Benefit                                              13

          Section 9.5. Interpretation                                       13

          Section 9.6. Governing Law                                        14



                              GUARANTEE AGREEMENT

     This GUARANTEE AGREEMENT, dated as of March 31, 1997, is executed and
delivered by GREATER BAY BANCORP, a California corporation (the "Guarantor")
having its principal office at 2860 West Bayshore Road, Palo Alto, California
94303, and WILMINGTON TRUST COMPANY a Delaware banking corporation, as trustee
(the "Guarantee Trustee"), for the benefit of the Holders from time to time of
the Preferred Securities (as defined herein) of GBB CAPITAL I, a Delaware
statutory business trust (the "Trust").

     WHEREAS, pursuant to an Amended and Restated Trust Agreement, dated as of
March 31, 1997 (the "Trust Agreement"), among the Guarantor, as Depositor,
Wilmington Trust Company as Property Trustee, Wilmington Trust Company, as
Delaware Trustee, and the Administrative Trustees named therein and the Holders
from time to time of undivided beneficial interests in the assets of the Trust,
the Trust issued $20,000,000 aggregate Liquidation Amount (as defined in the
Trust Agreement) of its 9.75% Cumulative Trust Preferred Securities, Liquidation
Amount $25 per Trust Preferred Security (the " Preferred Securities")
representing preferred undivided beneficial interests in the assets of the Trust
and having the terms set forth in the Trust Agreement;

     WHEREAS, the Preferred Securities will be issued by the Trust and the
proceeds thereof, together with the proceeds from the issuance of the Trust's
Common Securities (as defined below), will be used to purchase the Debentures
(as defined in the Trust Agreement) of the Guarantor which was deposited with
Wilmington Trust Company, as Property Trustee under the Trust Agreement, as
trust assets;

     WHEREAS, as an incentive for the Holders to purchase the Preferred
Securities, the Guarantor desires irrevocably and unconditionlly to agree, to
the extent set forth herein, to pay to the Holders of the Preferred Securities
the Guarantee Payments (as defined herein) and to make certain other payments on
the terms and conditions set forth herein.

     NOW, THEREFORE, in consideration of the purchase  by each Holder of
Preferred Securities, which purchase the Guarantor hereby agrees shall benefit
the Guarantor, the Guarantor executes and delivers this Guarantee Agreement and
pursuant to Section 5.1 hereof extends the Guarantee for the benefit of the
Holders from time to time of the Preferred Securities.

                            ARTICLE 1.  DEFINITIONS

     SECTION 1.1.  Definitions.

     As used in this Guarantee Agreement, the terms set forth below shall,
unless the context otherwise requires, have the following meanings.  Capitalized
or otherwise defined terms used but not otherwise defined herein shall have the
meanings assigned to such terms in the Trust Agreement and the Indenture (as
defined herein), each as in effect on the date hereof.

     "Affiliate" of any specified Person means any other Person directly or
indirectly controlling or controlled by or under direct or indirect common
control with such specified Person; provided, however, that an Affiliate of the
Guarantor shall not be deemed to be an Affiliate of the Trust.  For the purposes
of this definition, "control" when used with respect to any specified Person
means the power to direct the management and policies of such Person, directly
or indirectly, whether through the ownership of voting 

 
securities, by contract or otherwise; and the terms "controlling" and
"controlled" have meanings correlative to the foregoing.

     "Board of Directors" means either the board of directors of the Guarantor
or any committee of that board duly authorized to act hereunder.

     "Common Securities" means the securities representing common undivided
beneficial interests in the assets of the Trust.

     "Event of Default" means a default by the Guarantor on any of its payment
or other obligations under this Guarantee Agreement; provided, however, that,
except with respect to a default in payment of any Guarantee Payments, the
Guarantor shall have received notice of default and shall not have cured such
default within 60 days after receipt of such notice.

     "Guarantee"  has the meaning set forth in Section 5.1.

     "Guarantee Payments" means the following payments or distributions, without
duplication, with respect to the Preferred Securities, to the extent not paid or
made by or on behalf of the Trust:  (i) any accrued and unpaid Distributions (as
defined in the Trust Agreement) required to be paid on the Preferred Securities,
to the extent the Trust shall have funds on hand available therefor at such
time, (ii) the applicable Redemption Price (as defined in the Trust Agreement),
to the extent the Trust shall have funds on hand available therefor at such
time, and (iii) upon a voluntary or involuntary termination, winding up or
liquidation of the Trust, unless Debentures are distributed to the Holders, the
lesser of (a) the aggregate of the Liquidation Distribution (as defined in the
Trust Agreement) and (b) the amount of assets of the Trust remaining available
for distribution to Holders of Preferred Securities after satisfaction of
liabilities to creditors of the Trust as required by applicable law.

     "Guarantee Trustee" means Wilmington Trust Company, until a Successor
Guarantee Trustee has been appointed and has accepted such appointment pursuant
to the terms of this Guarantee Agreement, and thereafter means each such
Successor Guarantee Trustee.

     "Holder" means any holder, as registered on the books and records of the
Trust, of any Preferred Securities; provided, however, that in determining
whether the holders of the requisite percentage of Preferred Securities have
given any request, notice, consent or waiver hereunder, "Holder" shall not
include the Guarantor, the Guarantee Trustee, or any Affiliate of the Guarantor
or the Guarantee Trustee.

     "Indenture" means the Junior Subordinated Indenture dated as of March 31,
1997, as supplemented and amended between the Guarantor and Wilmington Trust
Company, as trustee.

     "List of Holders" has the meaning specified in Section 2.2(a).

     "Majority in Liquidation Amount of the Preferred Securities" means, except
as provided by the Trust Indenture Act, a vote by the Holder(s), voting
separately as a class, of more than 50% of the Liquidation Amount of all then
outstanding Preferred Securities issued by the Trust.

     "Officers' Certificate" means, with respect to any Person, a certificate
signed by the Chairman or a Vice Chairman of the Board of Directors of such
Person or the President or a Vice President of such Person, and by the
Treasurer, an Assistant Treasurer, the Secretary or an Assistant Secretary of
such Person, and delivered to the Guarantee Trustee. Any Officers' Certificate
delivered with respect to compliance with a condition or covenant provided for
in this Guarantee Agreement shall include:

     (a) a statement that each officer signing the Officers' Certificate has
read the covenant or condition and the definitions relating thereto;

 
     (b) a brief statement of the nature and scope of the examination or
investigation undertaken by each officer in rendering the Officers' Certificate;

     (c) a statement that each officer has made such examination or
investigation as, in such officer's opinion, is necessary to enable such officer
to express an informed opinion as to whether or not such covenant or condition
has been complied with; and

     (d) a statement as to whether, in the opinion of each officer, such
condition or covenant has been complied with.

     "Other Guarantees" means any guarantees similar to the Guarantee issued,
from time to time, by the Guarantor on behalf of holders of one or more series
of Preferred Securities issued by one or more GBB Trusts (as defined in the
Indenture) other than the Trust.

     "Person" means a legal person, including any individual, corporation,
estate, partnership, joint venture, association, joint stock company, limited
liability company, trust, unincorporated association, or government or any
agency or political subdivision thereof, or any other entity of whatever nature.

     "Responsible Officer" means, with respect to the Guarantee Trustee, any
officer of the Corporate Trust Department of the Guarantee Trustee and also
means, with respect to a particular corporate trust matter, any other officer to
whom such matter is referred because of that officer's knowledge of and
familiarity with the particular subject.

     "Securities Act" means the Securities Act of 1933, as amended.

     "Successor Guarantee Trustee" means a successor Guarantee Trustee
possessing the qualifications to act as Guarantee Trustee under Section 4.1.

     "Trust Indenture Act" means the Trust Indenture Act of 1939, as amended.

                       ARTICLE II.  TRUST INDENTURE ACT

     SECTION 2.1.  Trust Indenture Act; Application.

     (a) This Guarantee Agreement is subject to the provisions of the Trust
Indenture Act that are required to be part of this Guarantee Agreement and
shall, to the extent applicable, be governed by such provisions.

     (b) If and to the extent that any provision of this Guarantee Agreement
limits, qualifies or conflicts with the duties imposed by Sections 310 to 317,
inclusive, of the Trust Indenture Act, such imposed duties shall control.

     SECTION 2.2.  List of Holders.

     (a) The Guarantor shall furnish or cause to be furnished to the Guarantee
Trustee (a) semiannually, on or before January 15 and July 15 of each year, a
list, in such form as the Guarantee Trustee may reasonably require, of the names
and addresses of the Holders ("List of Holders") as of a date not more than 15
days prior to the delivery thereof, and (b) at such other times as the Guarantee
Trustee may request in writing, within 30 days after the receipt by the
Guarantor of any such request, a List of Holders as of a date not more than 15
days prior to the time such list is furnished, in each case to the extent such
information is in the possession or control of the Guarantor and is not
identical to a previously supplied list of Holders or 

                                      -3-

 
has not otherwise been received by the Guarantee Trustee in its capacity as
such. The Guarantee Trustee may destroy any List of Holders previously given to
it on receipt of a new List of Holders.

     (b) The Guarantee Trustee shall comply with its obligations under Section
311(a), Section 311(b) and Section 312(b) of the Trust Indenture Act.

     SECTION 2.3.  Reports by the Guarantee Trustee.

     Not later than July 15 of each year, commencing July 15, 1997, the
Guarantee Trustee shall provide to the Holders such reports as are required by
Section 313 of the Trust Indenture Act, if any, in the form and in the manner
provided by Section 313 of the Trust Indenture Act. The Guarantee Trustee shall
also comply with the requirements of Section 313(d) of the Trust Indenture Act.

     SECTION 2.4.  Periodic Reports to the Guarantee Trustee.

     The Guarantor shall provide to the Guarantee Trustee, the Securities and
Exchange Commission and the Holders such documents, reports and information, if
any, as required by Section 314 of the Trust Indenture Act and the compliance
certificate required by Section 314 of the Trust Indenture Act, in the form, in
the manner and at the times required by Section 314 of the Trust Indenture Act.

     SECTION 2.5.  Evidence of Compliance with Conditions Precedent.

     The Guarantor shall provide to the Guarantee Trustee, on an annual basis,
such evidence of compliance with such conditions precedent, if any, provided for
in this Guarantee Agreement that relate to any of the matters set forth in
Section 314(c) of the Trust Indenture Act. Any certificate or opinion required
to be given by an officer pursuant to Section 314(c)(1) may be given in the form
of an Officers' Certificate.

     SECTION 2.6.  Events of Default; Waiver.

     The Holders of a Majority in Liquidation Amount of the Preferred Securities
may, by vote, on behalf of the Holders, waive any past Event of Default and its
consequences. Upon such waiver, any such Event of Default shall cease to exist,
and any Event of Default arising therefrom shall be deemed to have been cured,
for every purpose of this Guarantee Agreement, but no such waiver shall extend
to any subsequent or other default or Event of Default or impair any right
consequent therefrom.

     SECTION 2.7.  Event of Default; Notice.

     (a) The Guarantee Trustee shall, within 90 days after the occurrence of an
Event of Default, transmit by mail, first class postage prepaid, to the Holders,
notices of all Events of Default known to the Guarantee Trustee, unless such
defaults have been cured before the giving of such notice, provided, that,
except in the case of a default in the payment of a Guarantee Payment, the
Guarantee Trustee shall be protected in withholding such notice if and so long
as the Board of Directors, the executive committee or a trust committee of
directors and/or Responsible Officers of the Guarantee Trustee in good faith
determines that the withholding of such notice is in the interests of the
Holders.

     (b) The Guarantee Trustee shall not be deemed to have knowledge of any
Event of Default unless the Guarantee Trustee shall have received written
notice, or a Responsible Officer charged with the administration of this
Guarantee Agreement shall have obtained written notice, of such Event of
Default.

     SECTION 2.8.  Conflicting Interests.

                                      -4-

 
     The Trust Agreement shall be deemed to be specifically described in this
Guarantee Agreement for the purposes of clause (i) of the first proviso
contained in Section 310(b) of the Trust Indenture Act.

        ARTICLE III.  POWERS, DUTIES AND RIGHTS OF THE GUARANTEE TRUSTEE

     SECTION 3.1.  Powers and Duties of the Guarantee Trustee.

     (a)  This Guarantee shall be held by the Guarantee Trustee for the benefit
of the Holders, and the Guarantee Trustee shall not transfer this Guarantee to
any Person except to a Holder exercising his or her rights pursuant to Section
5.4(iv) or to a Successor Guarantee Trustee on acceptance by such Successor
Guarantee Trustee of its appointment to act as Successor Guarantee Trustee. The
right, title and interest of the Guarantee Trustee shall automatically vest in
any Successor Guarantee Trustee, upon acceptance by such Successor Guarantee
Trustee of its appointment hereunder, and such vesting and cessation of title
shall be effective whether or not conveyancing documents have been executed and
delivered pursuant to the appointment of such Successor Guarantee Trustee.

     (b)  If an Event of Default has occurred and is continuing, the Guarantee
Trustee shall enforce this Guarantee for the benefit of the Holders.

     (c)  The Guarantee Trustee, before the occurrence of any Event of Default
and after the curing of all Events of Default that may have occurred, shall
undertake to perform only such duties as are specifically set forth in this
Guarantee Agreement, and no implied covenants shall be read into this Guarantee
Agreement against the Guarantee Trustee. In case an Event of Default has
occurred (that has not been cured or waived pursuant to Section 2.6), the
Guarantee Trustee shall exercise such of the rights and powers vested in it by
this Guarantee Agreement, and use the same degree of care and skill in its
exercise thereof, as a prudent person would exercise or use under the
circumstances in the conduct of his or her own affairs.

     (d)  No provision of this Guarantee Agreement shall be construed to relieve
the Guarantee Trustee from liability for its own negligent action, its own
negligent failure to act or its own willful misconduct, except that:

          (i) prior to the occurrence of any Event of Default and after the
     curing or waiving of all such Events of Default that may have occurred:

              (A) the duties and obligations of the Guarantee Trustee shall be
          determined solely by the express provisions of this Guarantee
          Agreement, and the Guarantee Trustee shall not be liable except for
          the performance of such duties and obligations as are specifically set
          forth in this Guarantee Agreement; and

              (B) in the absence of bad faith on the part of the Guarantee
          Trustee, the Guarantee Trustee may conclusively rely, as to the truth
          of the statements and the correctness of the opinions expressed
          therein, upon any certificates or opinions furnished to the Guarantee
          Trustee and conforming to the requirements of this Guarantee
          Agreement; but in the case of any such certificates or opinions that
          by any provision hereof or of the Trust Indenture Act are specifically
          required to be furnished to the Guarantee Trustee, the Guarantee
          Trustee shall be under a duty to examine the same to determine whether
          or not they conform to the requirements of this Guarantee Agreement;

          (ii) the Guarantee Trustee shall not be liable for any error of
     judgment made in good faith by a Responsible Officer of the Guarantee
     Trustee, unless it shall be proved that the Guarantee Trustee was negligent
     in ascertaining the pertinent facts upon which such judgment was made;

                                      -5-

 
          (iii) the Guarantee Trustee shall not be liable with respect to any
     action taken or omitted to be taken by it in good faith in accordance with
     the direction of the Holders of not less than a Majority in Liquidation
     Amount of the Preferred Securities relating to the time, method and place
     of conducting any proceeding for any remedy available to the Guarantee
     Trustee, or exercising any trust or power conferred upon the Guarantee
     Trustee under this Guarantee Agreement; and

          (iv) no provision of this Guarantee Agreement shall require the
     Guarantee Trustee to expend or risk its own funds or otherwise incur
     personal financial liability in the performance of any of its duties or in
     the exercise of any of its rights or powers, if the Guarantee Trustee shall
     have reasonable grounds for believing that the repayment of such funds or
     liability is not reasonably assured to it under the terms of this Guarantee
     Agreement or adequate indemnity against such risk or liability is not
     reasonably assured to it.

     SECTION 3.2.  Certain Rights of Guarantee Trustee.

     (a)  Subject to the provisions of Section 3.1:

          (i) The Guarantee Trustee may rely and shall be fully protected in
     acting or refraining from acting upon any resolution, certificate,
     statement, instrument, opinion, report, notice, request, direction,
     consent, order, bond, debenture, note, other evidence of indebtedness or
     other paper or document reasonably believed by it to be genuine and to have
     been signed, sent or presented by the proper party or parties.

          (ii) Any direction or act of the Guarantor contemplated by this
     Guarantee Agreement shall be sufficiently evidenced by an Officers'
     Certificate unless otherwise prescribed herein.

          (iii) Whenever, in the administration of this Guarantee Agreement, the
     Guarantee Trustee shall deem it desirable that a matter be proved or
     established before taking, suffering or omitting to take any action
     hereunder, the Guarantee Trustee (unless other evidence is herein
     specifically prescribed) may, in the absence of bad faith on its part,
     request and rely upon an Officers' Certificate which, upon receipt of such
     request from the Guarantee Trustee, shall be promptly delivered by the
     Guarantor.

          (iv) The Guarantee Trustee may consult with legal counsel, and the
     written advice or opinion of such legal counsel with respect to legal
     matters shall be full and complete authorization and protection in respect
     of any action taken, suffered or omitted to be taken by it hereunder in
     good faith and in accordance with such advice or opinion. Such legal
     counsel may be legal counsel to the Guarantor or any of its Affiliates and
     may be one of its employees. The Guarantee Trustee shall have the right at
     any time to seek instructions concerning the administration of this
     Guarantee Agreement from any court of competent jurisdiction.

          (v) The Guarantee Trustee shall be under no obligation to exercise any
     of the rights or powers vested in it by this Guarantee Agreement at the
     request or direction of any Holder, unless such Holder shall have provided
     to the Guarantee Trustee such adequate security and indemnity as would
     satisfy a reasonable person in the position of the Guarantee Trustee,
     against the costs, expenses (including attorneys' fees and expenses) and
     liabilities that might be incurred by it in complying with such request or
     direction, including such reasonable advances as may be requested by the
     Guarantee Trustee; provided that, nothing contained in this Section
     3.2(a)(v) shall be taken to relieve the Guarantee Trustee, upon the
     occurrence of an Event of Default, of its obligation to exercise the rights
     and powers vested in it by this Guarantee Agreement.

                                      -6-

 
          (vi) The Guarantee Trustee shall not be bound to make any
     investigation into the facts or matters stated in any resolution,
     certificate, statement, instrument, opinion, report, notice, request,
     direction, consent, order, bond, debenture, note, other evidence of
     indebtedness or other paper or document, but the Guarantee Trustee, in its
     discretion, may make such further inquiry or investigation into such facts
     or matters as it may see fit.

          (vii) The Guarantee Trustee may execute any of the trusts or powers
     hereunder or perform any duties hereunder either directly or by or through
     its agents or attorneys, and the Guarantee Trustee shall not be responsible
     for any misconduct or negligence on the part of any such agent or attorney
     appointed with due care by it hereunder.

          (viii) Whenever in the administration of this Guarantee Agreement the
     Guarantee Trustee shall deem it desirable to receive instructions with
     respect to enforcing any remedy or right or taking any other action
     hereunder, the Guarantee Trustee (A) may request instructions from the
     Holders, (B) may refrain from enforcing such remedy or right or taking such
     other action until such instructions are received, and (C) shall be
     protected in acting in accordance with such instructions.

     (b)  No provision of this Guarantee Agreement shall be deemed to impose any
duty or obligation on the Guarantee Trustee to perform any act or acts or
exercise any right, power, duty or obligation conferred or imposed on it in any
jurisdiction in which it shall be illegal, or in which the Guarantee Trustee
shall be unqualified or incompetent in accordance with applicable law, to
perform any such act or acts or to exercise any such right, power, duty or
obligation. No permissive power or authority available to the Guarantee Trustee
shall be construed to be a duty to act in accordance with such power and
authority.

     SECTION 3.3.  Indemnity.

     The Guarantor agrees to indemnify the Guarantee Trustee for, and to hold it
harmless against, any loss, liability or expense incurred without negligence or
bad faith on the part of the Guarantee Trustee, arising out of or in connection
with the acceptance or administration of this Guarantee Agreement, including the
costs and expenses of defending itself against any claim or liability in
connection with the exercise or performance of any of its powers or duties
hereunder.

                         ARTICLE IV.  GUARANTEE TRUSTEE

     SECTION 4.1.  Guarantee Trustee:  Eligibility.

     (a)  There shall at all times be a Guarantee Trustee which shall:

          (i) not be an Affiliate of the Guarantor; and

          (ii) be a Person that is eligible pursuant to the Trust Indenture Act
     to act as such and has a combined capital and surplus of at least
     $50,000,000, and shall be a corporation meeting the requirements of Section
     310(a) of the Trust Indenture Act. If such corporation publishes reports of
     condition at least annually, pursuant to law or to the requirements of the
     supervising or examining authority, then, for the purposes of this Section
     4.1(a)(ii) and to the extent permitted by the Trust Indenture Act, the
     combined capital and surplus of such corporation shall be deemed to be its
     combined capital and surplus as set forth in its most recent report of
     condition so published.

                                      -7-

 
     (b) If at any time the Guarantee Trustee shall cease to be eligible to so
act under Section 4.1(a), the Guarantee Trustee shall immediately resign in the
manner and with the effect set out in Section 4.2(c).

     (c) If the Guarantee Trustee has or shall acquire any "conflicting
interest" within the meaning of Section 310(b) of the Trust Indenture Act, the
Guarantee Trustee and Guarantor shall in all respects comply with the provisions
of Section 310(b) of the Trust Indenture Act.

     SECTION 4.2.  Appointment, Removal and Resignation of the Guarantee
                   Trustee.

     (a) Subject to Section 4.2(b), the Guarantee Trustee may be appointed or
removed without cause at any time by the Guarantor.

     (b) The Guarantee Trustee shall not be removed until a Successor Guarantee
Trustee has been appointed and has accepted such appointment by written
instrument executed by such Successor Guarantee Trustee and delivered to the
Guarantor.

     (c) The Guarantee Trustee appointed hereunder shall hold office until a
Successor Guarantee Trustee shall have been appointed or until its removal or
resignation.  The Guarantee Trustee may resign from office (without need for
prior or subsequent accounting) by an instrument in writing executed by the
Guarantee Trustee and delivered to the Guarantor, which resignation shall not
take effect until a Successor Guarantee Trustee has been appointed and has
accepted such appointment by instrument in writing executed by such Successor
Guarantee Trustee and delivered to the Guarantor and the resigning Guarantee
Trustee.

     (d) If no Successor Guarantee Trustee shall have been appointed and
accepted appointment as provided in this Section 4.2 within 60 days after
delivery to the Guarantor of an instrument of resignation, the resigning
Guarantee Trustee may petition, at the expense of the Guarantor, any court of
competent jurisdiction for appointment of a Successor Guarantee Trustee.  Such
court may thereupon, after prescribing such notice, if any, as it may deem
proper, appoint a Successor Guarantee Trustee.

                             ARTICLE V.  GUARANTEE

     SECTION 5.1.  Guarantee.

     The Guarantor irrevocably and unconditionally agrees to pay in full on a
subordinated basis to the Holders the Guarantee Payments (without duplication of
amounts theretofore paid by or on behalf of the Trust), as and when due,
regardless of any defense, right of set-off or counterclaim which the Trust may
have or assert other than the defense of payment (the "Guarantee").  The
Guarantee is a continuing guarantee, and the Guarantor fully, knowingly and
unconditionally waives any right the Guarantor may have to revoke the Guarantee
as to any future transactions under Section 2815 of the California Civil Code or
otherwise.  The Guarantor's obligation to make a Guarantee Payment may be
satisfied by direct payment of the required amounts by the Guarantor to the
Holders or by causing the Trust to pay such amounts to the Holders.

     SECTION 5.2.  Waiver of Notice and Demand.

     The Guarantor hereby waives notice of acceptance of the Guarantee and of
any liability to which it applies or may apply, presentment, demand for payment,
any right to require a proceeding first against the Guarantee Trustee, Trust or
any other Person before proceeding against the Guarantor, protest, notice of
nonpayment, notice of dishonor, notice of redemption and all other notices and
demands.

     SECTION 5.3.  Obligations Not Affected.

                                      -8-

 
     The obligations, covenants, agreements and duties of the Guarantor under
this Guarantee Agreement shall in no way be affected or impaired by reason of
the happening from time to time of any of the following:

     (a) the release or waiver, by operation of law or otherwise, of the
performance or observance by the Trust of any express or implied agreement,
covenant, term or condition relating to the Preferred Securities to be performed
or observed by the Trust;

     (b) the extension of time for the payment by the Trust of all or any
portion of the Distributions (other than an extension of time for payment of
Distributions that results from the extension of any interest payment period on
the Debentures as provided in the Indenture), Redemption Price, Liquidation
Distribution or any other sums payable under the terms of the Preferred
Securities or the extension of time for the performance of any other obligation
under, arising out of, or in connection with, the Preferred Securities;

     (c) any failure, omission, delay or lack of diligence on the part of the
Holders to enforce, assert or exercise any right, privilege, power or remedy
conferred on the Holders pursuant to the terms of the Preferred Securities, or
any action on the part of the Trust granting indulgence or extension of any
kind;

     (d) the voluntary or involuntary liquidation, dissolution, sale of any
collateral, receivership, insolvency, bankruptcy, assignment for the benefit of
creditors, reorganization, arrangement, composition or readjustment of debt of,
or other similar proceedings affecting, the Trust or any of the assets of the
Trust;

     (e) any invalidity of, or defect or deficiency in, the Preferred
Securities;

     (f) the settlement or compromise of any obligation guaranteed hereby or
hereby incurred; or

     (g) any other circumstance whatsoever that might otherwise constitute a
legal or equitable discharge or defense of a guarantor, it being the intent of
this Section 5.3 that the obligations of the Guarantor hereunder shall be
absolute and unconditional under any and all circumstances.

There shall be no obligation of the Holders to give notice to, or obtain the
consent of, the Guarantor with respect to the happening of any of the foregoing.

     SECTION 5.4.  Rights of Holders.

     The Guarantor expressly acknowledges that:  (i) this Guarantee will be
deposited with the Guarantee Trustee to be held for the benefit of the Holders;
(ii) the Guarantee Trustee has the right to enforce this Guarantee on behalf of
the Holders; (iii) the Holders of a Majority in Liquidation Amount of the
Preferred Securities have the right to direct the time, method and place of
conducting any proceeding for any remedy available to the Guarantee Trustee in
respect of this Guarantee Agreement or exercising any trust or power conferred
upon the Guarantee Trustee under this Guarantee Agreement; and (iv) any Holder
may institute a legal proceeding directly against the Guarantor to enforce its
rights under this Guarantee Agreement, without first instituting a legal
proceeding against the Guarantee Trustee, the Trust or any other Person.

     SECTION 5.5.  Guarantee of Payment.

     This Guarantee creates a guarantee of payment and not of collection.  This
Guarantee will not be discharged except by payment of the Guarantee Payments in
full (without duplication of amounts theretofore paid by the Trust) or upon
distribution of Debentures to Holders as provided in the Trust Agreement.

                                      -9-

 
     SECTION 5.6.  Subrogation.

     The Guarantor shall be subrogated to all (if any) rights of the Holders
against the Trust in respect of any amounts paid to the Holders by the Guarantor
under this Guarantee Agreement and shall have the right to waive payment by the
Trust pursuant to Section 5.1; provided, however, that the Guarantor shall not
(except to the extent required by mandatory provisions of law) be entitled to
enforce or exercise any rights which it may acquire by way of subrogation or any
indemnity, reimbursement or other agreement, in all cases as a result of payment
under this Guarantee, if, at the time of any such payment, any amounts are due
and unpaid under this Guarantee.  If any amount shall be paid to the Guarantor
in violation of the preceding sentence, the Guarantor agrees to hold such amount
in trust for the Holders and to pay over such amount to the Holders.

     SECTION 5.7.  Independent Obligations.

     The Guarantor acknowledges that its obligations hereunder are independent
of the obligations of the Trust with respect to the Preferred Securities and
that the Guarantor shall be liable as principal and as debtor hereunder to make
Guarantee Payments pursuant to the terms of this Guarantee Agreement
notwithstanding the occurrence of any event referred to in subsections (a)
through (g), inclusive, of Section 5.3 hereof.

                    ARTICLE VI.  COVENANTS AND SUBORDINATION

     SECTION 6.1.  Subordination.

     The obligations of the Guarantor under this Guarantee will constitute
unsecured obligations of the Guarantor and will rank subordinate and junior in
right of payment to all Senior and Subordinated Debt (as defined in the
Indenture) in the same manner as Debentures (as defined in the Trust Agreement).

     SECTION 6.2.  Pari Passu Guarantees.

     The obligations of the Guarantor under this Guarantee shall rank pari passu
with the obligations of the Guarantor under all Other Guarantees.


       ARTICLE VII.  CONSOLIDATION, MERGER, CONVEYANCE, TRANSFER OR LEASE

     SECTION 7.1.  Guarantor May Consolidate, Etc., Only on Certain Terms.

     The Guarantor shall not consolidate with or merge into any other Person or
convey, transfer or lease its properties and assets substantially as an entirety
to any Person, and no Person shall consolidate with or merge into the Guarantor
or convey, transfer or lease its properties and assets substantially as an
entirety to the Guarantor, unless:

     (1) in case the Guarantor shall consolidate with or merge into another
Person or convey, transfer or lease its properties and assets substantially as
an entirety to any Person, the Person formed by such consolidation or into which
the Guarantor is merged or the Person which acquires by conveyance or transfer,
or which leases, the properties and assets of the Guarantor substantially as an
entirety shall be a corporation, partnership or trust organized and existing
under the laws of the United States of America or any State or the District of
Columbia, and shall expressly assume the Guarantor's obligations under this
Guarantee;

     (2) immediately after giving effect thereto, no Event of Default, and no
event which, after notice or lapse of time, or both, would become an Event of
Default, shall have happened and be continuing;

                                     -10-

 
     (3) such consolidation, merger, conveyance, transfer or lease is permitted
under the Trust Agreement and the Indenture and does not give rise to any breach
or violation of the Trust Agreement or the Indenture; and

     (4) the Guarantor has delivered to the Guarantee Trustee an Officers'
Certificate and an Opinion of Counsel, each stating that such consolidation,
merger, conveyance, transfer or lease and assumption of the Guarantor's
obligations under this Guarantee Agreement comply with this Article and that all
conditions precedent herein provided for relating to such transaction have been
complied with; and the Guarantee Trustee, subject to Section 3.1 hereof, may
rely upon such Officers' Certificate and Opinion of Counsel as conclusive
evidence that such transaction complies with this Section 7.1.

     SECTION 7.2.  Successor Guarantor Substituted.

     Upon any consolidation or merger by the Guarantor with or into any other
Person, or any conveyance, transfer or lease by the Guarantor of its properties
and assets substantially as an entirety to any Person in accordance with Section
7.1, the successor Person formed by such consolidation or into which the
Guarantor is merged or to which such conveyance, transfer or lease is made shall
succeed to, and be substituted for, and may exercise every right and power of,
the Guarantor under this Guarantee Agreement with the same effect as if such
successor Person had been named as the Guarantor herein; and in the event of any
such conveyance, transfer or lease the Guarantor shall be discharged from all
obligations and covenants under this Guarantee Agreement.

                           ARTICLE VIII.  TERMINATION

     SECTION 8.1.  Termination.

     This Guarantee Agreement shall terminate and be of no further force and
effect upon the earliest of (i) full payment of the applicable Redemption Price
of all Preferred Securities, (ii) the distribution of Debentures to the Holders
in exchange for all of the Preferred Securities or (iii) full payment of the
amounts payable in accordance with the Trust Agreement upon liquidation of the
Trust. Notwithstanding the foregoing clauses (i) through (iii), this Guarantee
Agreement will continue to be effective or will be reinstated if it has been
terminated pursuant to one of such clauses (i) through (iii), as the case may
be, if at any time any Holder must restore payment of any sums paid with respect
to Preferred Securities or this Guarantee Agreement.

                          ARTICLE IX.  MISCELLANEOUS

     SECTION 9.1.  Successors and Assigns.

     All guarantees and agreements contained in this Guarantee Agreement shall
bind the successors, assigns, receivers, trustees and representatives of the
Guarantor and shall inure to the benefit of the Holders of the Preferred
Securities then outstanding.  Except in connection with a consolidation, merger
or sale involving the Guarantor that is permitted under Article VII hereof and
Article VIII of the Indenture, the Guarantor shall not assign its obligations
hereunder.

     SECTION 9.2.  Amendments.

     Except with respect to any changes which do not adversely affect the rights
of the Holders in any material respect (in which case no vote will be required),
this Guarantee Agreement may not be amended without the prior approval of the
Holders of not less than a Majority in Liquidation Amount of the Preferred

                                     -11-

 
Securities.  The provisions of Article VI of the Trust Agreement concerning
meetings of the Holders shall apply to the giving of such approval.

     SECTION 9.3.  Notices.

     Any notice, request or other communication required or permitted to be
given hereunder shall be in writing, duly signed by the party giving such
notice, and delivered, telecopied or mailed by first class mail as follows:

     (a)  if given to the Guarantor, to the address set forth below or such
other address, facsimile number or to the attention of such other Person as the
Guarantor may give notice to the Holders:

          Greater Bay Bancorp
          2860 West Bayshore Road
          Palo Alto, California 94303

          Facsimile No.:  (415) 323-7421
          Attention: David L. Kalkbrenner

     (b)  if given to the Trust, in care of the Guarantee Trustee, at the
Trust's (and the Guarantee Trustee's) address set forth below or such other
address as the Guarantee Trustee on behalf of the Trust may give notice to the
Holders:

          GBB Capital I
          c/o Greater Bay Bancorp
          2860 West Bayshore Road
          Palo Alto, California 94303

          Facsimile No.:  (415) 323-7421
          Attention: David L. Kalkbrenner

          with a copy to:
 
          Wilmington Trust Company
          1100 North Market
          Wilmington, Delaware  19890

          Facsimile No.: (302) 651-1000
          Attention:  Corporate Trust Administration

     (c)  if given to any Holder, at the address set forth on the books and
records of the Trust.

     All notices hereunder shall be deemed to have been given when received in
person, telecopied with receipt confirmed, or mailed by first class mail,
postage prepaid, except that if a notice or other document is refused delivery
or cannot be delivered because of a changed address of which no notice was
given, such notice or other document shall be deemed to have been delivered on
the date of such refusal or inability to deliver.

     SECTION 9.4.  Benefit.

                                     -12-

 
     This Guarantee is solely for the benefit of the Holders and is not
separately transferable from the Preferred Securities.

     SECTION 9.5.  Interpretation.

     In this Guarantee Agreement, unless the context otherwise requires:

     (a) capitalized terms used in this Guarantee Agreement but not defined in
the preamble hereto have the respective meanings assigned to them in Section
1.1;

     (b) a term defined anywhere in this Guarantee Agreement has the same
meaning throughout;

     (c) all references to "the Guarantee Agreement" or "this Guarantee
Agreement" are to this Guarantee Agreement as modified, supplemented or amended
from time to time;

     (d) all references in this Guarantee Agreement to Articles and Sections are
to Articles and Sections of this Guarantee Agreement unless otherwise specified;

     (e) a term defined in the Trust Indenture Act has the same meaning when
used in this Guarantee Agreement unless otherwise defined in this Guarantee
Agreement or unless the context otherwise requires;

     (f) a reference to the singular includes the plural and vice versa; and

     (g) the masculine, feminine or neuter genders used herein shall include the
masculine, feminine and neuter genders.

     SECTION 9.6.  Governing Law.

     THIS GUARANTEE AGREEMENT SHALL BE GOVERNED BY AND CONSTRUED AND INTERPRETED
IN ACCORDANCE WITH THE LAWS OF THE STATE OF CALIFORNIA WITHOUT REGARD TO THE
CONFLICT OF LAW PRINCIPLES THEREOF.

                                     -13-

 
     This instrument may be executed in any number of counterparts, each of
which so executed shall be deemed to be an original, but all such counterparts
shall together constitute but one and the same instrument.

     THIS GUARANTEE AGREEMENT is executed as of the day and year first above
written.

                                    GREATER BAY BANCORP

                                    By: /s/ Steven C. Smith
                                        ----------------------------
                                    Name:   Steven C. Smith
                                    Title:  E.V.P. C.O.O. and C.F.O.


                                    WILMINGTON TRUST COMPANY
                                    as Guarantee Trustee

                                    By: /s/ James Lawler
                                        ----------------------------
                                    Name:   James Lawler
                                    Title:  Vice President