A fuel cell system includes a fuel cell with a cathode space and an anode space that are separated by a polymer electrolyte membrane. The cathode space is supplied with an oxygen-containing gas, and the anode space is supplied with a hydrogen-containing gas. A gas-generation device is provided in which a hydrogen-rich reformate containing carbon monoxide, is produced from a fuel with the aid of steam reforming and/or partial oxidation. A gas-cleaning stage selectively oxidizes the carbon monoxide in the reformate on a suitable catalyst while oxygen is added; and a heat exchanger which is arranged in the gas-cleaning stage, has a cooling medium flowing through it and serves to dissipate thermal energy. According to the invention, to dissipate the thermal energy, the anode off-gas and/or cathode off-gas emerging from the fuel cell and/or a starting-material gas which is present in the fuel cell system flows through the heat exchanger.