Exhibit 5.1

(713) 758-1074                                                    (713) 615-5926

                                 May 15, 1998


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
("Halliburton"), in connection with Halliburton's registration statement on Form
S-4 (the "Registration Statement") relating to the offering, sale and delivery
of shares (the "Shares") of Halliburton's common stock, par value $2.50 per
share, pursuant to the proposed merger of Halliburton N.C., Inc., a Delaware
corporation and wholly owned subsidiary of Halliburton, with and into Dresser
Industries, Inc., a Delaware corporation. Capitalized terms used but not defined
in this opinion are defined in the Registration Statement and are used herein
with the same meanings as ascribed to them therein.

        Before rendering our opinion, we examined certain of the corporate 
records of Halliburton, including its Restated Certificate of Incorporation, 
certain resolutions of the Board of Directors of Halliburton, the Registration 
Statement and the exhibits thereto and such certificates of corporate officers
of Halliburton and governmental officials as we deemed necessary for the
purposes of this opinion. As to matters of fact relevant to the opinions
expressed herein, and as to factual matters arising in connection with our
examination of the above described documents, we relied upon certificates and
other communications of corporate officers of Halliburton and governmental
officials without further investigation as to the facts set forth therein.

        Based upon the foregoing, we are of the opinion that, subject to 
adoption of the Charter Amendment by the stockholders of Halliburton and the
filing of the Charter Amendment with the Secretary of the State of Delaware, the
Shares have been validly authorized for issuance and, upon issuance as described
in the Registration Statement, will be validly issued, fully paid and
nonassessable.

        We hereby consent to the filing of this letter 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 
Securities Act of 1933, as amended, or the rules and regulations of the 
Securities and Exchange Commission issued thereunder.  For purposes of the 
foregoing opinions,

 
Halliburton Company
Page 2
May 15, 1998

we assumed that the Shares will be offered, issued and sold in compliance with 
state securities or Blue Sky laws.


                                                Very truly yours,



                                                VINSON & ELKINS L.L.P.