Drive device for unwinding a reel of foil in an unwinding module of a stamping machine, having an inactive position, and an active position in engagement with the reel, and comprising: a belt, having an inactive configuration, and an active configuration for contacting a portion of the reel and driven by a driver of the module so as to unwind the reel, rollers, maintaining the belt in the inactive and the active configuration, a translatable roller, which engages the belt and compensates for variations in length when the belt moves from the inactive to the active configuration, and a biasing arrangement, attached to the roller and thereby keeping the belt tensioned in both the inactive and the active configuration, and the biasing arrangement is actuated and disengaged from the belt, to move the belt from the inactive to the active configuration and vice versa.