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                                                                   EXHIBIT 10.13

                             PURE RESOURCES, INC.
                               500 West Illinois
                             Midland, Texas 79701


                                  May 9, 2001


Union Oil Company of California
2141 Rosecrans Avenue, Suite 4000
El Segundo, California 90245
Attention: Mr. Terry Dallas, Chief Financial Officer

     Re:  Pure Resources, Inc. - Business Opportunities Agreement

Ladies and Gentlemen:

     Reference is made to the Business Opportunities Agreement (the "Agreement")
dated as of December 13, 1999 among Union Oil Company of California ("Union
Oil"), Pure Resources, Inc. (formerly named Titan Resources Holdings, Inc.) (the
"Company"), TRH, Inc., and Titan Exploration, Inc.  Capitalized terms used and
not otherwise defined herein shall have the meanings given to them in the
Agreement.  The Agreement was entered into to address certain legal requirements
and concerns as well as for certain practical business purposes.   Although this
letter does request a limited waiver of the Agreement, we wish to make it clear
that we understand that the Agreement will remain in effect after any grant of a
limited waiver and that we will continue to remain focused on the business we
conduct in the Designated Area and any other area that you may permit us to
operate in pursuant to this or any other limited waiver.

     Pursuant to Section 1 of the Agreement, the Company agreed that, except
with the consent of Union Oil (which it may withhold in its sole discretion),
neither the Company nor its subsidiaries will engage in any business other than
the E&P Business and none of them will pursue any business opportunity that
involves any direct or indirect ownership interest in any properties located
outside the Designated Area.

     On January 30, 2001, you granted the Company a limited waiver of the
Agreement in connection with a transaction that resulted in the Company's owning
certain of the Assets (as defined below) and a majority of the partnership
interests in a limited partnership (the "Partnership") that owns certain of the
Assets (as defined).  As used in this letter, "Assets" means certain oil and gas
fee mineral interests, royalty interests, oil and gas leasehold interests and
related contracts and assets located in Alabama, Arkansas, Florida, Louisiana,
Mississippi, Oklahoma, Texas and the offshore area of the Gulf of Mexico (with
limited holdings located in Michigan, New Mexico, New York, North Dakota,
Pennsylvania and Wyoming) previously owned by International Paper Company or its
affiliates (collectively, the "Assets").
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May 9, 2001
Page 2

     As contemplated at the time the January 30, 2001 limited waiver was given,
you and the Company have been engaged in discussions regarding a broader waiver.
In connection therewith, notwithstanding Section 1 or any other provision in the
Agreement, we hereby request your limited waiver and consent:

     (a)  to the Company's (and its subsidiaries') pursuing and engaging in the
          E&P Business on or within the continental United States (the "Onshore
          Extended Business"); and

     (b)  subject to the terms and conditions set forth below, to the Company's
          (and its subsidiaries') pursuing and engaging in the E&P Business
          within the colored areas indicated on the map attached to this letter
          as Annex A, including direct or indirect ownership interests in
          properties located within any of such areas (the "Offshore Extended
          Business"); provided, however, that the term Offshore Extended
          Business shall not include E&P Business that both (1) does not arise
          out of the ownership by the Company or one of its subsidiaries of
          partnership or other interests in a partnership or program identified
          on Annex B and (2) relates to a prospect with gross unrisked reserve
          target potential of less than 20 Bcf.

     By signing below, Union Oil hereby (1) evidences its limited waiver of the
application of the Agreement's restrictions to the Onshore Extended Business and
the Offshore Extended Business and (2) acknowledges that neither the Onshore
Extended Business nor the Offshore Extended Business shall be deemed a breach or
violation of the Agreement.  Notwithstanding the foregoing, the limited waiver
that Union Oil grants hereby with respect to the Offshore Extended Business
shall expire and terminate on March 31, 2003 if Union Oil notifies the Company
in writing at any time in the month of  September,  2002, that this limited
waiver shall so terminate and that Union Oil will not extend the effective
period of such limited waiver beyond March 31, 2003.  If Union Oil notifies the
Company in writing, during the month of September, 2002, that the limited waiver
for Offshore Extended Business shall be extended beyond March 31, 2003, such
limited waiver will be so extended for the period of time set forth in Union
Oil's notification.   Union Oil's right to so notify the Company to either
terminate or extend the limited waiver for Offshore Extended Business may be
exercised in its sole discretion.

     We hereby acknowledge that if you grant the limited waiver requested above
or any more restrictive waiver, that such limited waiver will only apply to the
Onshore Extended Business and the Offshore Extended Business and in no way shall
serve to waive any provision of the Agreement that may be applicable to future
transactions or investments by the Company that are not included within the
Onshore Extended Business or the Offshore Extended Business.  Furthermore,
nothing herein shall serve as a general waiver or be construed as amending or
revising in any way the definition of "Designated Area" in the Agreement or any
other provision of the Agreement.
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May 9, 2001
Page 3

     By signing below, Union Oil also agrees that the provisions of the fifth
paragraph (regarding a right of first refusal) of that certain letter agreement
dated January 30, 2001 between the Company and Union Oil, which letter agreement
contained a limited waiver of the Agreement, are hereby terminated and of no
force or effect.

     As consideration for Union Oil's limited waiver as contemplated by this
letter and the agreement in the preceding paragraph, the Company, on behalf of
itself and its subsidiaries, subject to Union Oil's execution of this letter and
a Mineral and Royalty Deed (the "Mineral Deed") in the form attached hereto as
Annex C (or in such other form as is reasonably acceptable to the Company),
hereby sells, assigns, conveys, transfers and delivers, as effective as of 7:00
a.m. local time on April 1, 2001 (the "Effective Time"), to Union Oil,  all
right, title and interest as described in the Mineral Deed of the Company (or
its subsidiaries) in and to all the oil, gas and other minerals in and under and
that may be produced from the lands identified in the Mineral Deed and/or
Exhibit A thereto (the "Lands"), subject to any existing leases or contracts
currently in existence which affect the Lands, without warranty of title of any
kind whatsoever.  Union Oil agrees, upon such conveyance, to assume and
indemnify the Company (or its applicable subsidiary) for costs and liabilities
relating to the ownership of the subject Lands in accordance with the terms set
forth in the Mineral Deed.  The transactions described in this paragraph are
referred to below as the "Transfer."

     Provided that the Transfer occurs, appropriate adjustments shall be made
between Union Oil and the Company (or its subsidiary) so that (i) Union Oil will
receive all proceeds from sales of hydrocarbons (together with their constituent
products and any other minerals produced in association therewith, "Oil and
Gas") that are produced and saved from and after the Effective Time, net of all
applicable production, severance, and similar taxes, and net of all costs and
expenses that are incurred in the ownership or administration of the Lands from
and after the Effective Time and (ii) the Company (or its subsidiary) will
receive all proceeds from sales of Oil and Gas that are produced and saved for
the period of the Company's (or it's subsidiary's) ownership of the Lands prior
to the Effective Time, net of all applicable production, severance, and similar
taxes, and net of all costs and expenses that are incurred in the ownership,
operation or administration of the Lands prior to the Effective Time.

     The documents to effect the Transfer shall be in such forms as are
reasonably acceptable to Union Oil and the Company.  At the request of either
party hereto and without further consideration, the other party hereto shall
execute and deliver to such requesting party such instruments and documents and
take such other action (but without incurring any material financial obligation)
as such requesting party may reasonably request in order to consummate fully and
effectively the Transfer.

     We hereby confirm that the Transfer has been approved by a majority of the
independent/outside directors of the Company.
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May 9, 2001
Page 4

     By signing below, you agree that nothing in this letter terminates or
revokes any prior waiver of the Agreement previously granted by you.

     Please sign this letter in the space below, fax a signed copy to Joe
Dannenmaier of  Thompson & Knight L.L.P. (fax 214/ 969-1751) and return the
signed original to Pure Resources, Inc. c/o Joe Dannenmaier, Thompson & Knight
L.L.P., 1700 Pacific Avenue, Suite 3300, Dallas, Texas  75201.

                                    Very truly yours,

                                    PURE RESOURCES, INC.


                                    By:  /s/ Jack Hightower
                                      ---------------------------------------
                                        Jack D. Hightower,
                                        President and Chief Executive Officer

Accepted and agreed to:


UNION OIL COMPANY OF CALIFORNIA


By: /s/ Terry Dallas
    -----------------------------------
    Name:   Terry G. Dallas
    Title:  Executive Vice President
            and Chief Financial Officer
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May 9, 2001
Page 5

                                    Annex B


1.  T3D Partnership.  Texas 3D Ventures, L.P., formed pursuant to Agreement of
    ---------------
    Limited Partnership entered into as of April 1, 1995 by and among Houston
    Energy, Inc. (formerly Houston Energy & Development, Inc.), as general
    partner, and IP Petroleum Company, Inc. et al, as limited partners, as
    amended.

2.  T3D Program.  Program Agreement dated effective April 1, 1995 by and between
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    Texas 3D Ventures, L.P. and IP Petroleum Company, Inc., et al, as amended
    and restated effective October 1, 1999.

3.  LOV Program.  Program Agreement dated effective January 1, 1994, by and
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    between Louisiana Offshore Ventures II and IP Petroleum Company, Inc., et
    al, as amended and restated effective October 1, 1999.

4.  El-Can Program.  Agreement dated effective May 1, 2000, by and between Andex
    --------------
    Resources, L.L.C., El-Can Exploration, Inc. and IP Petroleum, Inc.