An injection-molding press for overmolding a second part onto a first part (8) includes a movable portion and a fixed portion, with a first mold element (11, 12) on one of the press portions and a second mold element (20) on the other press portion so as to form a cavity (C) between the mold portions when the press is closed. The first mold portion (11, 12) has a first support surface (125a) movable in the first mold portion and the second mold portion (20) has a second support surface (22a). In the closed position of the press with a first part (8) in the first mold portion and the first support surface (125a) advanced in the working position, at least a portion (8b) of the front end of the first part (8) is clamped between the two support surfaces (125a, 22a), forming a tight seal for the cavity (C).