A mechanical press is adapted to perform a hydroforming operation. The press includes a stationary base supporting a bolster that carries a lower die section, a movable slide that carries an upper die section, and a drive mechanism for selectively raising and lowering the slide relative to the base. An inflatable bladder is disposed between the base and the bolster. Initially, the drive mechanism is operated to lower the slide to its bottom dead center position such that the die sections enclose a workpiece within a die cavity. Then, pressurized fluid is supplied within the workpiece to expand it outwardly into conformance with the die cavity. At the same time, pressurized fluid is also supplied to the interior of the inflatable bladder. The application of such pressurized fluid causes the inflatable bladder to be physically expanded, causing the bolster carrying the lower die section to be raised relative to the base. The magnitude of this force is preferably selected to be approximately equal to the magnitude of the outwardly directed force exerted by the workpiece against the lower die section and, thus, the bolster. As a result, the lower die section is urged upwardly to remain in position relative to the upper die section during the hydroforming operation. In effect, the inflatable bladder pre-stresses the frame of the press and fills any extra space created by the deflections of the various components of the frame, thereby retaining the die sections and in position during the hydroforming operation. As a result, the press can be adapted to hydroform relatively large and thick-walled workpieces, such as vehicle frame components.