The Born This Way musician, who met Kinney while filming the song's official promo, is seen wearing a wedding dress and kissing her besuited boyfriend in the piece.
The clip also features GaGa writhing in torn fishnet stockings and a leather strap outfit, as well as scenes of Kinney posing shirtless.
GaGa previously posted a picture of a naked pool kiss with the Vampire Diaries actor to her new social networking site Little Monsters.
GaGa and Kinney were first romantically linked back in September 2011, but separated in May for about a month before reuniting.
The couple were the subject of false engagement rumours earlier this year when GaGa was photographed by Terry Richardson wearing a new ring.