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                                                                     EXHIBIT 5.1


                       [BAKER & BOTTS, L.L.P. LETTERHEAD]

                                                                  March 23, 1999


Conoco Inc.
600 North Dairy Ashford
Houston, Texas 77079

Gentlemen:

         As set forth in the Registration Statement on Form S-3 (Registration
No. 333-72291), as amended (the "Registration Statement"), filed with the
Securities and Exchange Commission (the "Commission") by Conoco Inc., a Delaware
corporation (the "Company"), under the Securities Act of 1933, as amended (the
"Act"), relating to unsecured debt securities of the Company (the "Securities")
to be issued and sold by the Company from time to time pursuant to Rule 415
under the Act for an aggregate initial offering price not to exceed
$4,000,000,000, certain legal matters in connection with the Securities are
being passed upon for you by us.

         In our capacity as your counsel in the connection referred to above, we
have examined (i) the Second Amended and Restated Certificate of Incorporation
of the Company and the By-Laws of the Company, each as amended to date
(together, the "Charter Documents"), (ii) the form of the Indenture filed as
Exhibit 4.1 to the Registration Statement to be executed by the Company and the
trustee thereunder (the "Indenture"), pursuant to which the Securities may be
issued, and (iii) the originals, or copies certified or otherwise identified, of
corporate records of the Company, certificates of public officials and of
representatives of the Company, statutes and other instruments and documents as
a basis for the opinions hereafter expressed.

         In connection with this opinion, we have assumed that (i) the
Registration Statement and any amendments thereto (including post-effective
amendments) will have become effective; (ii) a prospectus supplement will have
been prepared and filed with the Commission describing the Securities offered
thereby; (iii) all Securities will be issued and sold in compliance with
applicable federal and state securities laws and in the manner stated in the
Registration Statement and the appropriate prospectus supplement; (iv) a
definitive purchase, underwriting or similar agreement with respect to any
Securities offered will have been duly authorized and validly executed and
delivered by the Company and the other parties thereto; and (v) any securities
issuable upon conversion, redemption, exchange or exercise of any Securities
being offered will be duly authorized, created and, if appropriate, reserved for
issuance upon such conversion, redemption, exchange or exercise.


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Conoco Inc.                           -2-                         March 23, 1999


         Based upon and subject to the foregoing, we are of the opinion that:

         1. The Company is a corporation duly organized and validly existing in
     good standing under the laws of the State of Delaware.

         2. When (i) the Indenture has been validly executed and delivered by
     the Company to the trustee; (ii) the Indenture has been duly qualified
     under the Trust Indenture Act of 1939, as amended; (iii) the Board of
     Directors of the Company or, to the extent permitted by the General
     Corporation Law of the State of Delaware and the Company's Charter
     Documents, a duly constituted and acting committee thereof (such Board of
     Directors or committee being hereinafter referred to as the "Board") has
     taken all necessary corporate action to approve and establish the terms of
     the Securities, to approve the issuance thereof and the terms of the
     offering thereof and related matters; and (iv) such Securities have been
     duly executed, authenticated, issued and delivered in accordance with both
     the provisions of the Indenture and either (a) the provisions of the
     applicable definitive purchase, underwriting or similar agreement approved
     by the Board upon payment of the consideration therefor provided for
     therein or (b) upon conversion, redemption, exchange or exercise of any
     other Security, in accordance with the terms of such Security or the
     instrument governing such Security providing for such conversion,
     redemption, exchange or exercise as approved by the Board, for the
     consideration approved by the Board, such Securities will constitute legal,
     valid and binding obligations of the Company, enforceable against the
     Company, except as the enforceability thereof is subject to the effect of
     (i) bankruptcy, insolvency, reorganization, moratorium, fraudulent
     conveyance or other laws relating to or affecting creditors' rights
     generally and (ii) general principles of equity (regardless of whether such
     enforceability is considered in a proceeding in equity or at law).

         We hereby consent to the filing of this opinion of counsel as Exhibit
5.1 to the Registration Statement. We also consent to the reference to our Firm
under the heading "Legal Opinions" in the prospectus forming a part of the
Registration Statement. In giving this consent, we do not hereby admit that we
are in the category of persons whose consent is required under Section 7 of the
Act or the rules and regulations of the Commission thereunder.

                                      Very truly yours,

                                      BAKER & BOTTS, L.L.P.


JDK/KBR