Amendment No. 1 to Rights Agreement

Exhibit 10.6

AMENDMENT NO. 1 TO RIGHTS AGREEMENT

      This Amendment No. 1 to Rights Agreement (this “Amendment”) is dated and effective as of March 30, 2005, by and between Horizon Offshore, Inc., a Delaware corporation (the “Company”), and Mellon Investor Services, LLC., as Rights Agent (“Mellon”).

W I T N E S S E T H:

      WHEREAS, the Company entered into that certain Rights Agreement, dated January 11, 2002 (the “Rights Agreement”), by and between the Company and Mellon, as Rights Agent;

      WHEREAS, on March 29, 2005, the Board of Directors of the Company authorized the Company to enter into a Restructuring Letter Agreement (the “Restructuring Agreement”) with and the holders of the Company’s 16% and 18% Subordinated Secured Notes due March 31, 2007 (the “Subordinated Notes”) and outstanding shares of Company’s Series A Redeemable Participating Preferred Stock, $1.00 per value per share (the “Preferred Stock”), pursuant to which the Subordinated Notes and shares of Preferred Stock will be exchanged for the Company’s common stock and mandatorily convertible redeemable preferred stock; and

      WHEREAS, pursuant to Section 27 of the Rights Agreement, the Company and the Rights Agent desire to amend the definition of Acquiring Person in Section 1 of the Rights Agreement to ensure that the transactions consummated pursuant to the Restructuring Agreement do not cause a Triggering Event under the Rights Agreement;

      NOW, THEREFORE, in consideration of the premises and the mutual agreements set forth herein and in the Rights Agreement, the parties hereby agree as follows:

      1. Capitalized terms used but not defined herein shall have the meanings assigned to such terms in the Rights Agreement.

      2. The definition of Acquiring Person in Section 1 of the Rights Agreement is hereby amended, such that, as amended, it shall read in its entirety as follows:

      “Acquiring Person” means any Person who, together with all Affiliates and Associates of such Person, is the Beneficial Owner of the Threshold Percentage or more of the shares of Common Stock then outstanding, but shall not include an Exempt Person; provided, however, that (a) if the Board determines in good faith that a Person who would otherwise be an Acquiring Person has become the Beneficial Owner of a number of shares of Common Stock such that the Person would otherwise qualify as an Acquiring Person inadvertently (including, without limitation, because (i) such Person was unaware that it beneficially owned a percentage of Common Stock that would otherwise cause such Person to be an

Acquiring Person or (ii) such Person was aware of the extent of its Beneficial Ownership of Common Stock but had no actual knowledge of the consequences of such Beneficial Ownership under this Agreement) and without any intention of changing, exercising or influencing control of the Company, then such Person shall not be deemed to be or to have become an Acquiring Person for any purposes of this Agreement unless and until such Person shall have failed to divest itself as soon as practicable (as determined, in good faith, by the Board of Directors of the Company) of Beneficial Ownership of a sufficient number of shares of Common Stock so that such Person would no longer otherwise qualify as an Acquiring Person; and (b) no Person shall become an Acquiring Person as the result of any acquisition of shares of Common Stock by the Company which, by reducing the number of shares of Common Stock outstanding, increases the proportionate number of shares of Common Stock beneficially owned by such Person to the Threshold Percentage or more of the shares of Common Stock then outstanding; provided, however, that if a Person shall become the Beneficial Owner of the Threshold Percentage or more of the shares of Common Stock then outstanding by reason of such share acquisition by the Company and shall thereafter become the Beneficial Owner of any additional shares of Common Stock (other than pursuant to a dividend or distribution paid or made by the Company on the outstanding Common Stock or pursuant to a split or subdivision of the outstanding Common Stock), then such Person shall be deemed to be an Acquiring Person unless upon becoming the Beneficial Owner of such additional shares of Common Stock such Person does not beneficially own the Threshold Percentage or more of the shares of Common Stock then outstanding. Notwithstanding anything contained herein to the contrary, no party to the Restructuring Agreement or any of such party’s respective Affiliates or Associates shall be deemed to be an Acquiring Person.

      3. Except as amended hereby, the Rights Agreement and all exhibits thereto shall remain in full force and effect.

      4. This Amendment may be executed in multiple counterparts and each such counterpart shall for all purposes be deemed an original, and all such counterparts shall together constitute one and the same instrument.

      IN WITNESS WHEREOF, the parties hereto have caused this Amendment to be executed in their names and on their behalf by and through their duly authorized officers, effective as of the date first above written.

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