A cylindrical heat exchanger for use as a gas cooler in a thermal regenerative machine such as a Stirling engine includes an imperforate middle wall of sufficient strength and thickness to withstand the pressure exerted by the working fluid. The heat exchanger includes an inner corrugated wall located within an axial gas flow passage inside the middle wall, and an outer corrugated wall which defines an axial coolant flow passage along the outer surface of the middle wall. The coolant flow passage preferably contains a corrugated intermediate wall.