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Hyundai Samho Heavy Industries is said to have inked an order for two 9,000-teu containerships from a Singapore-based shipowner.
Gulf Copper will work on the rig in its Galveston yard before sending it off to work for Mexico's state-run Pemex.
The 83 is a Marathon LeTourneau Class 82 SD-C jack-up that can drill in water depths up to 250 feet (76.2 metres) and well depths up to 25,000 feet.
It is the sixth Ensco rig Gulf Copper has worked on since Galveston was hit by Hurricane Ike, the company said in a release.