The deal requires the company to install flowlines, steel catenary risers, umbilicals, jumpers and related subsea architecture.
Subsea 7 will tie-back two drill centres to a tension leg platform, hosting drill centres, water injection and gas lifts.
Production will go via two 10in flowlines from each drill centre.
The main offshore installation phase is anticipated to start in the third quarter of 2016, with completion expected in the first quarter of 2017.
Subsea 7 Gulf of Mexico vice-president Craig Broussard said: "Having worked with Hess before in the Gulf of Mexico on the Tubular Bells construction scope, as well as on flexible product installation for Hess' Ceiba Field in Equatorial Guinea, we are pleased to be continuing our work with this important client."
Chevron subsidiary Union Oil Company of California has a 25% working interest in the Stampede development, while Hess, Statoil and Nexen, each have 25% stakes.
The Stampede field has estimated recoverable resources of more than 300 million barrels of oil equivalent.
Production from the project is expected to be approximately 80,000 barrels of crude oil per day.
Hess will operate the Stampede Project, which includes the joint development of the Knotty Head and the Pony discoveries in Green Canyon blocks 511, 512, and 468.