An apparatus for detaching a supporting fabric from a substantially-elongated piece of elastomeric material includes at least one plane for supporting the piece, first rollers located close to an exit edge of the plane, at least one second roller, and at least one actuator for moving the second roller. The second roller moves between a first position, not in proximity to the first rollers, and a second position, in proximity to the first rollers. In the second position, the first rollers and the second roller detach the fabric from the piece.