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                                                                    EXHIBIT 10.3



              THIRD AMENDMENT TO THE REGISTRATION RIGHTS AGREEMENT


         This Third Amendment to the Registration Rights Agreement (the
"Amendment") is made and entered into this 30th day of March, 2000, by and among
Energy Capital Investment Company PLC, an English investment company ("Energy
PLC"), EnCap Energy Capital Fund III-B, L.P., a Texas limited partnership
("EnCap III-B"), BOCP Energy Partners, L.P., a Texas limited partnership
("BOCP"), EnCap Energy Capital Fund III, L.P., a Texas limited partnership
("EnCap III"), Kayne Anderson Energy Fund, L.P., a Delaware limited partnership
("Kayne"), BancAmerica Capital Investors SBIC I, L.P., a Delaware limited
partnership ("BACI"), Eos Partners, L.P., a Delaware limited partnership ("Eos
Partners"), Eos Partners SBIC, L.P., a Delaware limited partnership ("Eos
SBIC"), Eos Partners SBIC II, L.P., a Delaware limited partnership ("Eos SBIC
II" and together with Eos Partners and Eos SBIC, collectively referred to as
"EOS"), SGC Partners II LLC, a Delaware limited liability company ("SGCP") and
Banker's Trust Company, a New York banking corporation ("BT") (each
individually, an "Investor" and collectively, the "Investors"), EnCap Equity
1994 Limited Partnership, a Texas limited partnership ("EnCap LP") and Bargo
Energy Company, a Texas corporation (the "Company"), and evidences the
following:

                                    RECITALS

         A. The Company (as successor by merger to Future Petroleum Corporation,
a Utah corporation), Energy PLC and EnCap LP entered into a Registration Rights
Agreement on August 14,1998, as amended by a First Amendment to the Registration
Rights Agreement dated December 15, 1998 and as Amended by the Second Amendment
to the Registration Rights Agreement dated May 14, 1999 (as amended, the
"Agreement"), covering the shares of Common Stock issued to Energy PLC and EnCap
LP and the New Common Shares issued to the Investors;

         B. The Company and the Investors (other than BT) have entered into a
Subscription Agreement dated March 30, 2000 ("Subscription Agreement"), pursuant
to which the Investors have subscribed to purchase up to _____ shares of Common
Stock ("Subscription Common Shares")

         C. The Company, BT and certain other lenders have entered into a Credit
Agreement, dated March 30, 2000, pursuant to which, under the circumstances set
forth therein, certain loans made by BT may be converted into equity securities,
including up to shares of Common Stock ("Conversion Common Shares " and together
with the Subscription Common Shares, the "Additional Common Shares");

         D. The parties to the Agreement desire to amend the Agreement to cover
the Additional Common Shares, to add BT as a party to the Agreement and to make
certain other changes.



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                                    AGREEMENT

         NOW, THEREFORE, for and in consideration of the foregoing Recitals and
mutual covenants contained herein, the sufficiency of which is hereby
acknowledged, the parties hereto, intending to be legally bound, do hereby agree
as follows:

         Section 1 AMENDMENTS TO THE AGREEMENT. Section 1(a) of the Agreement is
amended as follows:

         (a) Clause (i) of the first line in the definition of "Registrable
Securities" shall be replaced with:

         "(i) the Fund I Shares, the New Common Shares, the Subscription Common
Shares and the Conversion Common Shares and"

         (b) There shall be added to Section 1(a) a definition of BT,
"Conversion Common Shares" and "Subscription Common Shares" as follows:

         "BT" means Bankers Trust Company, a New York banking corporation.

         "Conversion Common Shares" shall mean all of the shares of Common Stock
issued by the Company pursuant to the conversion of Tranche B Term Loans as
defined under the Credit Agreement, dated March 30, 2000, among the Company, BT
and the other lenders party thereto.

         "'Subscription Common Shares' shall mean all of the shares of Common
Stock issued by the Company pursuant to that certain Subscription Agreement,
dated March 30, 2000, by and among the Company, Energy PLC, EnCap III-B, BOCP,
EnCap III, Kayne, BACI, Eos Partners, Eos SBIC,, Eos SBIC II, and SGCP."

         Section 2 BINDING EFFECT. Each Investor, EnCap LP and the Company by
execution of this Agreement shall be bound by and subject to the terms and
obligations under the Agreement, as amended by this Amendment.

         Section 3 BT AS A PARTY. By execution of this Agreement, BT agrees to
be bound by and subject to the terms and obligations under the Agreement, as
amended by this Agreement, as if an original party thereto, from and after the
date on which it shall acquire any BT Shares and for so long as it shall hold
the same.

         Section 4 NO OTHER CHANGES. Except as explicitly amended by the
Amendment, the terms, conditions, rights and obligations under the Agreement
shall remain in full force and effect.

         Section 5 COUNTERPARTS. This Amendment may be executed by the parties
hereto in any number of counterparts, each of which shall be deemed an original,
but all of which shall constitute one and the same agreement.

         Section 6 NOTICES. All notices given to BT under this Amendment or the
Agreement shall be given in this matter provided in the Credit Agreement.


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         IN WITNESS WHEREOF, the Parties hereto have executed this Agreement as
of the date above first written.




                              ENERGY CAPITAL INVESTMENT COMPANY PLC



                              By:
                                 -------------------------------------------
                                 Gary R. Petersen
                                 Director

                              ENCAP ENERGY CAPITAL FUND III, L.P.
                              By: EnCap Investments L.L.C., General Partner



                              By:
                                 -------------------------------------------
                                 D. Martin Phillips
                                 Managing Director

                              ENCAP EQUITY 1994 LIMITED PARTNERSHIP
                              By: EnCap Investments L.L.C., General Partner



                              By:
                                 -------------------------------------------
                                 D. Martin Phillips
                                 Managing Director

                              ENCAP ENERGY CAPITAL FUND III-B, L.P.
                              By: EnCap Investments L.L.C., General Partner



                              By:
                                 -------------------------------------------
                                 D. Martin Phillips
                                 Managing Director




<PAGE>   4


                           BOCP ENERGY PARTNERS, L.P.
                           By: EnCap Investments L.L.C., Manager

                           By:
                              -------------------------------------------
                              D. Martin Phillips
                              Managing Director

                           EOS PARTNERS, L.P.



                           By:
                              -------------------------------------------
                              Brian D. Young
                              General Partner


                           EOS PARTNERS SBIC, L.P.
                           By: Eos SBIC General, L.P., its general partner
                               By: Eos SBIC, Inc., its general partner



                           By:
                              -------------------------------------------
                              Brian D. Young
                              President
                           EOS PARTNERS SBIC II, L.P.
                           By: Eos SBIC General II, L.P., its general partner
                               By: Eos SBIC II, Inc., its general partner



                           By:
                              -------------------------------------------
                              Brian D. Young
                              President




<PAGE>   5





          SGC PARTNERS II LLC



          By:
             -------------------------------------------
             V. Frank Pottow
             Managing Director

          BANCAMERICA CAPITAL INVESTORS SBIC I, L.P.
          By: BancAmerica Capital Management SBIC I, LLC, its general partner
              By: BancAmerica Capital Management I, L.P., its sole member
                  By: BACM I GP, LLC, its general partner



          By:
             -------------------------------------------
             J. Travis Hain
             Managing Director

          KAYNE ANDERSON ENERGY FUND, L.P.
          By: Kayne Anderson Capital Advisors, L.P., its General Partner
              By: Kayne Anderson Investment Management, Inc., its
                  General Partner



          By:
             -------------------------------------------
             Daniel M. Weingeist
             Managing Director

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                                BARGO ENERGY COMPANY



                                By:
                                   -------------------------------------------
                                   Jonathan M. Clarkson
                                   President



                                BANKERS TRUST COMPANY



                                By:
                                   -------------------------------------------
                                   Name:
                                   Title: