Exhibit 31.1 

**CertificationPursuant to Rule 13a-14(a) (17 CFR 240.13a-14(a))**

**or Rule 15d-14(a)(17 CFR 240.15d-14(a)) under the Securities Exchange Act of 1934**

I, Bill J. Lam, certify that: 

1.        I havereviewed this Annual Report on Form 10-K/A for the year 2003 of HorizonOffshore, Inc.; 

2.       Based on my knowledge, this report does not contain any untrue statement of amaterial fact or omit to state a materialfact necessary to make the statementsmade, in light of the circumstances under which such statements were made, notmisleading with respect to the period covered by this report; 

3.       Based on my knowledge, the financial statements, and other financial informationincluded in this report, fairly present in all material respects the financialcondition, results of operations and cash flows of the registrant as of, andfor, the periods presented in this report; 

4.       The registrant’s other certifying officer(s) and I are responsible forestablishing and maintaining disclosure controls and procedures (as defined inExchange Act Rules 13a-15(e) and 15d-15(e)) for the registrant and have: 

|   | a.     | Designed    such disclosure controls and procedures, or caused such disclosure controls    and    procedures to be designed under our supervision, to    ensure that material information relating to the registrant, including its    consolidated subsidiaries, is made known to us by others within those    entities, particularly during the period in which this report is being    prepared; |
|--------|--------|--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
|   |   |                                                                                                                                                                                                                                                                                                                                                                                                               |
|   | b.     | Evaluated    the effectiveness of the registrant's disclosure controls and procedures and    presented in this report our conclusions about the effectiveness of the    disclosure controls and procedures, as of the end of the period covered by    this report based on such evaluation; and                                                                                                                    |
|   |   |                                                                                                                                                                                                                                                                                                                                                                                                               |
|   | c.     | Disclosed    in this report any change in the registrant's internal control over    financial reporting that occurred during the registrant’s most recent fiscal    quarter (the registrant’s fourth fiscal quarter in the case of an annual    report) that has materially affected, or is reasonably likely to materially    affect, the registrant’s internal control over financial reporting.  |

5.       The registrant’s other certifying officer(s) and I have disclosed, based on ourmost recent evaluation of internal control over financial reporting, to theregistrant’s auditors and the audit committee of the registrant’s board ofdirectors (or persons performing the equivalent functions): 

|   | a.     | All    significant deficiencies and material weaknesses in the design or operation    of internal control over financial reporting which are reasonably likely to    adversely affect the registrant's ability to record, process, summarize and    report financial information; and |
|--------|--------|---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
|   |   |                                                                                                                                                                                                                                                                                  |
|   | b.     | Any fraud,    whether or not material, that involves management or other employees who    have a significant role in the registrant’s internal control over financial    reporting.                                                                                              |

Dated: April 8, 2004 

|                                                                                              | /s/ Bill J. Lam |
|---------------------------------------------------------------------------------------------------|-----------------|
|  
Bill J. Lam 

President and Chief Executive Officer

(Principal Executive Officer)

  |