Abstract: A device for exchanging substances, in particular, an oxygenator or dialyzer, comprises a stack of semipermeable membranes each having a plurality of longitudinally oriented capillaries embedded therein which open at each end. A porous tissue mass is disposed between adjacent membranes and define transverse gas flow paths therethrough from one side of said stack to the opposite side. A non-porous member closes off the top and the bottom of the stack and each longitudinal end of the tissue mass between adjacent membranes. A connecting piece is arranged at each end of the stack to provide for a flow path, for example, for blood, into the capillaries of each membrane and a similar connecting piece is arranged at the opposite end for the off-flow of the blood. Connecting pieces are arranged at each side for the inflow of a gas, for example, oxygen, into the tissues and for the outflow of the gas on the opposite side of the stack.