"A lot of great players have won here, so that's why we're happy to have our names on the cup. We are very proud," Gasquet said.
Mladenovic, however, failed to wrap up victory as she struggled to cope with the powerful groundstrokes of Vandeweghe, who brought the U.S. level by beating the Frenchwoman 6-4 7-5.
Vandeweghe's win forced the final to a deciding mixed doubles match and France won it 4-1 4-3(0) after a second-set tiebreak to deny the Americans a record seventh Hopman Cup title.
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"We came up short today. It's my fault. I'm sorry," Sock said.
France's only previous success in the mixed team event came in 2014.
(Reporting by Shravanth Vijayakumar in Bengaluru, editing by Ed Osmond)
