A bipolar cooling plate module used in fuel cell stacks contains an anode-side terminal plate and a cathode-side terminal plate, in which at least one of the surfaces of the two terminal plates facing towards one another contains coolant channels. The two terminal plates are fixed to one another by an integral sealing element. The sealing element contains a sealing bead that surrounds the coolant distribution structure and seals the gap between the cooling surfaces of the two terminal plates. A compressible two-dimensional conducting element is inserted between the terminal plates, which extends only over the region of the cooling surfaces enclosed by the sealing groove, in which the coolant distribution structure is located. The element has a compression reserve that is sufficient to compensate for the shrinkage in thickness caused by cooling, of the sealing bead.