**Exhibit 31.1****Section 302 Certification**I, David W. Sharp, certify that:     1. I have reviewed this quarterly report of Form 10-Qfor the quarter ended June 30, 2005 of Horizon Offshore,Inc.;     2. Based on my knowledge, this report does not contain anyuntrue statement of a material fact or omit to state a materialfact necessary to make the statements made, in light of thecircumstances under which such statements were made, notmisleading with respect to the period covered by this report;     3. Based on my knowledge, the financial statements, andother financial information included in this report, fairlypresent in all material respects the financial condition,results of operations and cash flows of the registrant as of,and for, the periods presented in this report;     4. The registrant’s other certifying officer(s) and Iare responsible for establishing and maintaining disclosurecontrols and procedures (as defined in Exchange ActRules 13a-15(e) and 15d-15(e)) and internal control overfinancial reporting (as defined in Exchange ActRules 13a-15(f) and 15d-15(f)) 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) Designed such internal control over financial    reporting, or caused such internal control over financial    reporting to be designed under our supervision, to provide    reasonable assurance regarding the reliability of financial    reporting and the preparation of financial statements for    external purposes in accordance with generally accepted    accounting principles;           (c) 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           (d) 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; and     5. The registrant’s other certifying officer(s) and Ihave disclosed, based on our most recent evaluation of internalcontrol over financial reporting, to the registrant’sauditors and the audit committee of the registrant’s boardof directors (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.     /s/ David W. Sharp       David W. Sharp     President and Chief Executive Officer     Horizon Offshore, Inc.Date: August 15, 2005