[LETTERHEAD OF VINSON & ELKINS APPEARS HERE]

                                 August 1, 1997

                                                                     EXHIBIT 5.1

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

Re:  Halliburton Company;  Debt Securities

Ladies and Gentlemen:

     We are acting as counsel to Halliburton Company, a Delaware corporation
(the "Company"), in connection with the offering, sale and delivery by it from
time to time, in one or more series, of its debt securities (the "Debt
Securities") with an aggregate initial offering price not to exceed $600,000,000
on terms to be determined at the time of offering.  The offering, sale and
delivery of the Debt Securities are being registered pursuant to the
registration provisions of the Securities Act of 1933, as amended, by virtue of
the Company's Registration Statement on Form S-3 to which this opinion is an
exhibit. The Debt Securities are to be issued pursuant to the Second Senior
Indenture dated as of December 1, 1996 between the Company and Texas Commerce
Bank National Association, as trustee, as amended, supplemented and modified by
the First and Second Supplemental Indentures dated as of December 5, 1996 and
December 12, 1996, respectively (the "Second Senior Indenture"), or the
Subordinated Indenture dated as of January 2, 1991 between the predecessor of
the Company and Texas Commerce Bank National Association, as trustee, as
amended, supplemented and modified by the First Supplemental Indenture dated as
of December 12, 1996 among the Company, the Company's predecessor and the
trustee (the "Subordinated Indenture"). The Second Senior Indenture and the
Subordinated Indenture are herein referred to as the "Indentures". Capitalized
terms used but not defined herein are defined in the Indentures and are used
herein with the same meanings as ascribed to them therein.

     In our capacity as counsel to the Company, we examined the Registration
Statement, the Prospectus and the Indentures. We also examined the certificate
of incorporation and by-laws of the Company and the minutes of all meetings and
all unanimous consents of the board of directors of the Company and any
committees thereof relating to the authorization, issuance and sale of the
Debt Securities. In addition, we examined and relied upon certificates and
telegrams of public officials and certificates of officers of the Company and we
made such other investigations and examined such other documents as we deemed
necessary as a basis for the opinions hereinafter expressed. In the

 
course of the foregoing investigations and examinations, we assumed the
genuineness of all signatures on, and the authenticity of, all documents and
instruments submitted to us as originals, the conformity to original documents
of all documents submitted to us as copies and the due authorization, execution
and delivery by the parties thereto of all documents and instruments examined by
us.

     Based on the foregoing and subject to the qualifications set forth below,
we are of the opinion that:

     (i)    The Company has been duly incorporated and is validly existing as a
corporation in good standing under the laws of the State of Delaware.
 
     (ii)   The Indentures have been duly authorized, executed and delivered by
the Company and (assuming each Indenture has been duly authorized, executed and
delivered by the Trustee) constitute legal, valid and binding agreements of the
Company, enforceable in accordance with their terms, except as enforcement
thereof may limited by bankruptcy, insolvency, fraudulent conveyance,
reorganization, moratorium and other similar laws relating to or affecting
enforcement of creditors' rights generally, general equitable principles
(whether considered in a proceeding in equity or at law) and an implied covenant
of good faith and fair dealing.

     (iii)  The Debt Securities, upon determination of the terms and provisions
thereof pursuant to a Terms Certificate issued by the Company to the Trustee,
will be duly authorized for issuance, and, when issued, authenticated and
delivered pursuant to the provisions of the Indentures against payment of the
consideration therefor, will constitute valid and legally binding obligations of
the Company, enforceable in accordance with their terms, except as enforcement
thereof may limited by bankruptcy, insolvency, fraudulent conveyance,
reorganization, moratorium and other similar laws relating to or affecting
enforcement of creditors' rights generally, general equitable principles
(whether considered in a proceeding in equity or at law) and an implied covenant
of good faith and fair dealing.
 
     The opinions herein expressed are limited to the federal laws of the United
States, the laws of the States of Texas and New York and the General Corporation
Law of the State of Delaware, all as in effect on the date hereof.

     We hereby consent to the reference to our Firm under the caption "Legal
Opinions" in the Prospectus constituting a part of the Registration Statement.

                                         Very truly yours,



                                         Vinson & Elkins L.L.P.