A process for drying a gas stream having a periodically fluctuating moisture content which assures that a predetermined dew point temperature is not exceeded at any time by conducting the gas stream through a moisture adsorbing medium having a reversible moisture adsorption capacity is greater than the average moisture content of the gas stream, and an apparatus for carrying out this process comprising a source of a flowing gas, such as air, which has a fluctuating moisture content, connected to a gas utilizing device via a chamber filled with an adsorbing medium which adsorbs moisture in a reversible manner and which has a moisture adsorption capacity which is greater than the average moisture content of the gas stream.