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                                  EXHIBIT 5.1


                       VINSON & ELKINS L.L.P. LETTERHEAD

                                October 7, 1997



Halliburton Company
3600 Lincoln Plaza
500 North Akard Street
Dallas, Texas  75201-3391

Ladies and Gentlemen:

     We acted as counsel for Halliburton Company, a Delaware corporation (the
"Company"), in connection with the Company's Registration Statement on Form S-8
(the "Registration Statement") relating to the offering and sale of up to
866,741 shares (the "Shares") of common stock, par value $2.50 per share, of
the Company pursuant to the plans set forth on the cover of the Registration
Statement (collectively, the "Plans").

     Before rendering our opinion, we examined the Registration Statement, the
Restated Certificate of Incorporation and the By-Laws, as amended, of the
Company and certain resolutions of the Board of Directors of the Company.

     Based upon the foregoing, we are of the opinion that the Shares to be
issued upon the exercise of stock options granted pursuant to the Plans have
been validly authorized for issuance and when the Registration Statement has
become effective under the Securities Act of 1933, as amended (the "Act"), and
the Shares are issued and paid for in accordance with the terms of the
appropriate Plan and any underlying stock option agreement, the Shares so issued
will be validly issued, fully paid and nonassessable.

     We hereby consent to the filing of this opinion as an exhibit to the
Registration Statement.  By giving such consent, we do not admit that we are
within the category of persons whose consent is required under Section 7 of the
Act or the rules and regulations of the Securities and Exchange Commission
issued thereunder.  For purposes of this opinion, we assume that the securities
to be issued pursuant to the Registration Statement will be issued in compliance
with all applicable state securities or Blue Sky laws.

                                    Very truly yours,


                                    /s/ VINSON & ELKINS L.L.P.
