An improved air-to-air, heat-and-moisture exchanger is described which facilitates the simultaneous transfer of both heat and water vapor between two streams of air while greatly inhibiting the transfer of pollutant gases. A rigid encasement supports a stack of flat, thin-film composite membranes that are held in parallel, spaced-apart relation, thereby forming a box-shaped exchanger. The membranes are synthesized by interfacial polymerization, a process whereby a thin, non-porous film may be uniformly deposited onto a porous substrate material. The thin film is hydrophilic so as to provide high water vapor flux together with excellent selectivity against the transfer of other gases and vapors. The substrate provides mechanical support. Each of the parallel sheets of composite membrane in the stack separates two continuously flowing streams of air, one comprised of the air being exhausted from a building and the other comprised of the fresh air brought in to replace the exhausted air, thereby providing simultaneous heat transfer and moisture transfer between the streams while selectively restricting the transfer of the pollutant gases found in the exhaust air.