First Amendment to Limited Liability Agreement of

EXHIBIT 10.4

First Amendment to Limited Liability Agreement

of

Contango Offshore Exploration LLC

and

Additional Agreements

This First Amendment to Limited Liability Company Agreement (this “Amendment”) of Contango Offshore Exploration LLC, a Delaware limited liability company (the “Company”), is entered into as of September 1, 2005, by and among COE Offshore, LLC, a Delaware limited liability company (“COE O/S”), Centaurus Oil and Gas, LP, a Texas limited partnership (“Centaurus”), and Juneau Exploration, L.P., a Texas limited partnership (“JEX”).

1. JEX has transferred portions of its interests in the Company to COE O/S and Centaurus. COE O/S consents to those transfers on the terms hereinafter set forth.

2. By signing this Amendment, Centaurus executes the Agreement, hereby acknowledging acceptance by Centaurus of all of the terms and provisions of the Agreement.

3. Section 5.A of the limited liability agreement of the Company dated November 1, 2002, as said agreement may have been amended (the “Agreement”), is hereby amended effective as of the date hereof by adding the following after the last sentence thereof:

“Notwithstanding the foregoing, Centaurus Oil and Gas, LP (“CENTAURUS”) shall have no obligation to contribute any amounts to the Company for working capital, and JEX shall have no right to demand any contribution from CENTAURUS.”

4. Section 5.B of the Agreement is hereby amended effective as of the date hereof by adding the following after the last sentence thereof:

“Notwithstanding the foregoing, CENTAURUS shall have no obligation to pay any amounts due under or in connection with the DATA or LICENSES.”

5. Section 8.A of the Agreement is hereby amended effective as of the date hereof in its entirety to read as follows:

“A. The Members will have the following interests in the profits and losses of the Company.

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6. Section 12.B of the Agreement is hereby amended effective as of the date hereof in its entirety to read as follows:

“B. While the COMPANY is in existence and for a period of one (1) year thereafter, COE O/S, Contango Oil & Gas Company, a Delaware corporation (“CONTANGO”), and CENTAURUS will not – and each of CONTANGO and CENTAURUS will cause any of its affiliates not to – without the approval of the other Members, (i) acquire any interest in any area covered by any seismic data utilized hereunder, or (ii) exploit, participate in the recovery or exploitation of, derive any benefit from, or assist anyone else to recover or exploit or derive any benefit from any area covered by any DATA utilized hereunder; provided, however, that CENTAURUS and its affiliates shall only be subject to this Section 12.B with respect to any DATA which CENTAURUS or any of its affiliates gain knowledge as a result of CENTAURUS’ membership in the COMPANY.”

7. In accordance with Section 8.D of the Agreement, it is hereby agreed that the Company will file an election under Section 754 of the Internal Revenue Code on its return for the current fiscal year.

8. In accordance with Section 13.C(ii) of the Agreement, the transfers of portions of the interest in the Company held by Juneau Exploration, L.P. that result in the interests in profits and losses of the Company set forth in Section 8.A of the Agreement, as amended above, are hereby approved.

9. Centaurus may transfer its entire interest under the Agreement to a newly-formed limited liability company or limited partnership of which Centaurus is the sole member or sole manager or sole general partner or sole limited partner, as the case may be, provided that:

(a) if the transferee is not a limited liability company of which Centaurus is the sole member, Centaurus certifies to the Company and the other parties hereto that the transfer was made in exchange for fair consideration or that the investors in Centaurus approved the transfer;

(b) if the transferee is not a limited liability company of which Centaurus is the sole member or a limited partnership of which Centaurus is the sole general partner, Centaurus certifies to the Company and the other parties hereto that the transferee is controlled (as defined below) by Centaurus or by an entity that Centaurus controls or by an entity that controls Centaurus; and

(c) the transferee signs the Agreement by executing an instrument as required by Section 13.D of the Agreement.

Control means the ability to elect a majority of the directors or members of the governing body of an entity or in any other manner to control or determine the management of an entity. Control may be direct or indirect through one or more entities.

Centaurus will indemnify each of the other parties hereto against any liability and hold each of them harmless from and pay any loss, damage, cost or expense (including, without

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limitation, legal fees and court costs) that any such party incurs on account of any claim against it arising out of or in connection with a transfer by Centaurus of its interest in the Company.

10. As hereinabove amended, the Agreement will remain in full force and effect.

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