1. Molding unit (10) for the forming of a container made of thermoplastic material by blow molding, particularly by stretch blow molding, of a preform, the molding unit (10) having:two half-molds (14) that are transversely mobile in relation to a fixed holder (26) between a molding joined position and a separated position for extraction of the finished container;
a mold bottom (16) that is vertically mobile between a lower position for extraction of the finished container and an upper position for molding of a base of the container, the mold bottom (16) being able to be moved between the two half-molds (14) in the molding joined position over a directed path, termed boxing, that extends from an initial intermediate boxing position to its upper molding position;
a carriage (24) that carries the mold bottom (16) and that is mounted to move in the fixed holder (26);
first actuation means to move the carriage (24) relative to its fixed holder (26) between a lower position of the mold bottom (16) and its initial intermediate boxing position;
second actuation means having at least one cylinder (32) housing a piston (98) sliding vertically that is able to push the mold bottom (16) over its boxing path by pressurizing a lower work chamber (100) of the cylinder (32);
wherein the mold bottom (16) is supported in a fixed manner by the carriage (24), the cylinder (32) being fixed relative to the fixed holder (26).