EXHIBIT 99.1
                                                                    ------------

                     Press Release dated August 13, 1998

 
                For Immediate Release For Information Contact
                ---------------------------------------------

August 13, 1998                          David L. Kalkbrenner, President & CEO
                                            (650) 614-5767
                                         Steven C. Smith, EVP, COO & CFO
                                            (650) 813-8222

                         GREATER BAY BANCORP ISSUES
                                 $30 MILLION
                        OF TRUST PREFERRED SECURITIES

     PALO ALTO, Calif. -- Greater Bay Bancorp (Nasdaq--"GBBK"), announced today
that it has completed an offering in the aggregate amount of $30 million through
GBB Capital II, a trust affiliate of the Company formed for the purpose of the
offering.  The securities issued in the offering were sold in a private
transaction pursuant to an applicable exemption from registration under the
Securities Act of 1933 (the "Act") and have not been registered under the Act.
The securities may not be offered or sold in the United States absent
registration or an applicable exemption from registration requirements. The
securities have an offering price (liquidation amount) of $1,000 per security
and will receive distributions at a variable rate of interest, initially at
7.1875%.  The interest rate will reset quarterly, equal to 3-month LIBOR plus
150 basis points, will be cumulative and will be payable quarterly.  As part of
the transaction, Greater Bay Bancorp negotiated an interest rate swap to fix the
cost of the offering at 7.55% for 10 years.
 
     GBB Capital II used the proceeds from the sale of the Floating Rate Trust
Preferred Securities to purchase Junior Subordinated Debentures of Greater Bay
Bancorp.  Greater Bay Bancorp intends to invest approximately $10.0 million of
the net proceeds in one or more of the Company's subsidiary banks to increase
their capital levels and intends to use the remaining net proceeds for general
corporate purposes.  Under applicable regulatory guidelines, Greater Bay Bancorp
expects that a certain portion of the Floating Rate Trust Preferred Securities
will qualify as Tier I Capital, and the remaining portion will qualify as Tier
II Capital.
 
     This press release shall not constitute an offer to sell or the
solicitation of an offer to buy nor shall there by any sale of these securities
in any state in which such offer, solicitation or sale would be unlawful prior
to registration or qualification under the securities law of any such state.
 
     This document may contain forward-looking statements that are subject to
risks and uncertainties that could cause actual results to differ materially
from those projected.  For a discussion of factors that could cause actual
results to differ, please see the Company's publicly available Securities and
Exchange Commission filings, including its annual report on Form 10K dated
December 31, 1997, and particularly the discussion of risk factors within that
document.

                        "WE INVEST IN RELATIONSHIPS"