An apparatus for packaging a product with an extensible film, including a conveyor for conveying a product to the apparatus, an evacuating device for evacuating a product from the apparatus at the end of packaging, an unwinding device to unwind the extensible film from a reel mounted on the apparatus, a tensioning device for tensioning the extensible film, the unwinding device including a first gripping element and a second gripping element that are movable independently of one another in an unwinding direction for unwinding the extensible film from the reel.