Toshiba said in a statement it will pay Total $815 million to take over the business, which the Japanese company previously said may cause losses of as much as 1 trillion yen ($9 billion).
The acquisition comes amid a drive by Total to expand its LNG portfolio, and follows its proposed $8.8-billion deal to acquire Anadarko's Africa assets, which include an LNG project in Mozambique.
Toshiba expects to complete the transaction by the March-end of the current financial year, following regulatory approval, and to book a 93 billion yen ($859 million) loss related to the deal.
Toshiba was locked into a contract to pay a fixed processing fee for LNG over 20 years from Freeport LNG - regardless of whether it could later find buyers for the fuel at prevailing rates.
China's ENN Ecological Holdings Co last year agreed to buy the business, but scrapped the pact in April citing a failure to get approvals from shareholders and a US panel that monitors foreign investments.
At the time analysts said a recent plunge in spot LNG prices had made the deal unattractive.