An apparatus for packaging a product comprises a packaging assembly to fix a film sheet to a support. The packaging assembly includes a lower tool having seats for receiving the support and an upper tool facing the lower tool and comprising a film holding plate for holding the film sheet. The film holding plate has an active surface for receiving the film sheet, where the upper and lower tools cooperating to define a packaging chamber. The packaging assembly is open to receive the film sheet in a first operating condition and is hermetically closed in a second operating condition. The film holding plate has a lateral surface extending substantially perpendicular to a plane defined by the active surface, the film holding plate comprising ejectors arranged in the lateral surface for ejecting a stream of gas substantially parallel to said plane and substantially away from a center of the active surface.