<PAGE>
 
THIS NOTE AND ANY INDEBTEDNESS EVIDENCED HEREBY IS SUBORDINATED TO ANY
INDEBTEDNESS, WHETHER NOW EXISTING OF HEREAFTER INCURRED,  OF THE MAKER HEREOF
UNDER THAT CERTAIN AMENDED AND RESTATED CREDIT AGREEMENT, DATED MAY 13, 1998,
AMONG HORIZON VESSELS, INC., HORIZON OFFSHORE CONTRACTORS, INC., HORIZON
OFFSHORE, INC. AND DEN NORSKE BANK ASA.


                          __________________________

                             HORIZON OFFSHORE, INC.

                                PROMISSORY NOTE

                               DUE JUNE 30, 2003

                          __________________________

$10,000,000.00                                                    Houston, Texas
                                                                  August 5, 1998

     FOR VALUE RECEIVED, Horizon Offshore, Inc., a Delaware corporation (the
"Maker"), promises to pay to the order of Highwood Partners, L.P., a Delaware
limited partnership ("Payee"), as provided below, at 712 Fifth Avenue, New York,
New York  10019 (or at such other address as any holders hereof may designate in
writing), the sum of ten million and no/100 dollars ($10,000,000.00), or such
lesser amount as may have been advanced by the Payee hereunder, together with
interest on the unpaid principal balance at the rate set forth below, on the
earlier to occur of June 30, 2003 or such earlier date as such amount may become
due and payable in accordance with the terms hereof (the "Maturity Date").
Payment for all amounts due hereunder shall be made in lawful money of the
United States of America paid at the address of the Payee, or in such other form
or by such other means as the Maker and any holder shall agree.

     1.   ADVANCEMENT OF FUNDS.  Horizon may from time to time request advances
pursuant to this Note, provided that no event has occurred which would
constitute an event of default hereunder, by presenting to Highwood a request
for borrowing (a "Request").  Within three days of the receipt of such Request,
Highwood shall advance the requested funds to such account as Horizon may direct
by wire transfer of immediately available funds.  Any amounts advanced hereunder
shall only be used to repurchase shares of Horizon's common stock, $1.00 par
value per share, in open market purchases or privately negotiated transactions
effected in accordance with Rule 10b-18 under the Securities Exchange Act of
1934, as amended.

     2.   PAYMENT.  The unpaid principal balance of this Note shall be paid on
the Maturity Date.

     The outstanding principal amount of this Note from time to time shall bear
interest until paid at the rate set forth below.  Interest shall be calculated
on a 360 day per year basis and paid for the 

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actual number of days elapsed (including the first but excluding the last day)
during any period and shall be paid quarterly on March 31, June 30, September 30
and December 31 of each year, commencing September 30, 1998, and with a final
interest payment in the amount of all outstanding interest then unpaid being due
on the Maturity Date. The rate of interest to be charged from time to time on
this Note shall be equal to, for any particular period, the rate of interest
payable during such period on amounts borrowed under that certain Amended and
Restated Credit Agreement, dated May 13, 1998 among Horizon Vessels, Inc. and
Horizon Offshore Contractors, Inc., as borrowers, the Maker as guarantor and Den
norske Bank ASA, or any amendment, renewal or extension thereof (the "Credit
Facility"). If at any time during the term of this Note the Credit Facility is
no longer in effect, outstanding balances under this Note shall bear interest at
a rate of 9 1/2% per annum.

     If a payment of principal or interest falls due on a Saturday, Sunday, or
any other day on which financial institutions are generally not open for
business in Houston, Texas, payment shall be made on the next preceding business
day.

     Any amount paid pursuant to Section 3 shall be applied first to any accrued
and unpaid interest then accrued, if any, and, then to the principal amounts due
hereunder.

     3.   PREPAYMENT.  The Maker may at any time prepay any amount outstanding
under this Note in whole or in part without notice or penalty.

     4.   DEFAULT.

          4.1  Events of Default.  The following events or actions shall
constitute an event of default ("Events of Default"):

               (a) Maker shall default in the payment when due of any principal
     or interest under the Note and such default shall continue for three (3)
     business days after such amount is due; or

               (b) Maker shall be in default in the performance of or compliance
     with any term of any evidence of any indebtedness in an aggregate
     outstanding principal amount in excess of $100,000 or of any mortgage,
     indenture or other agreement relating thereto or any other condition
     exists, and as a consequence of such default or condition such indebtedness
     has become, or has been declared due and payable prior to its stated
     maturity or before its regularly scheduled dates of payment; or

               (c) Maker shall (i) apply for or consent to the appointment of,
     or the taking of possession by, a receiver, custodian, trustee or
     liquidator of itself or of all or a substantial part of its property, (ii)
     make a general assignment for the benefit of its creditors, (iii) commence
     a voluntary case under the U.S. Bankruptcy Code (as now or hereafter in
     effect), (iv) file a petition seeking to take advantage of any other law
     relating to bankruptcy, insolvency, reorganization, winding-up, or
     composition or readjustment of debts, (v) fail to controvert in a timely
     and appropriate manner, or acquiesce in writing to, any petition filed
     against it in an involuntary case under the U.S. Bankruptcy Code, or (vi)
     take any corporate action for the purpose of effecting any of the
     foregoing; or

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               (d) A proceeding or case shall be commenced, without the
     application or consent of Maker in any court of competent jurisdiction,
     seeking (i) its liquidation, reorganization, dissolution or winding-up, or
     the composition or readjustment of its debts, (ii) the appointment of a
     trustee, receiver, custodian, liquidator or the like of Maker or of all or
     any substantial part of the assets of any of them, or (iii) similar relief
     in respect or Maker under any law relating to bankruptcy, insolvency,
     reorganization, winding-up, or composition or adjustment of debts, and such
     proceeding or case shall continue undismissed, or an order, judgment or
     decree approving or ordering any of the foregoing shall be entered and
     continue unstayed and in effect, for a period of sixty (60) days; or an
     order for relief against Maker shall be entered in an involuntary case
     under the U.S. Bankruptcy Code.

          4.2  Remedies Upon Default.  If any Event of Default shall occur, the
holder of the Note may declare the principal amount of the Note to be
immediately due and payable.  The holder of such accelerated Note may proceed to
protect and enforce its rights under the Note by a suit in equity, action at law
or other appropriate proceeding whether for the specific performance of any
agreements contained herein or for an injunction against a violation of any of
the terms or provisions hereof; or in aid of the exercise of any power granted
hereby or at law.  No right shall operate as a waiver thereof or otherwise
prejudice the rights of such holder and no consent or waiver shall extend beyond
the particular case and purpose involved.  No remedy conferred hereby shall be
exclusive of any other remedy referred to herein or now or hereafter available
at law, in equity, by statute or otherwise.

     5.   WAIVER PROVISIONS/COSTS.  Maker waives presentment for payment, notice
of nonpayment, notice of intent to accelerate and notice of acceleration,
protest, notice of protest, notice of dishonor or nonpayment, bringing of suit,
diligence in taking any action to collect amounts called for hereunder.

     6.   CHOICE OF LAW.  THIS NOTE HAS BEEN EXECUTED AND DELIVERED IN AND IS
INTENDED TO BE PERFORMED IN THE STATE OF TEXAS, AND SHALL BE CONSTRUED AND
ENFORCED IN ACCORDANCE WITH THE INTERNAL LAWS OF SUCH STATE.

     IN WITNESS WHEREOF, the Maker has caused this Note to be issued this 5th
day of August, 1998.

                              HORIZON OFFSHORE, INC.


                              By: /s/ Bill J. Lam
                                  -------------------------------
                                  Bill J. Lam
                                  President

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