A fuel cell system includes a gas generation device for the catalytic steam reforming of a steam/fuel mixture and/or for the partial oxidation of an oxygen/fuel mixture for generating a hydrogen-rich medium and a gas cleaning stage for removing carbon monoxide from the product gas of the gas generation device using selective CO oxidation. A principal flow path of the medium is split at least over a certain length into a first and a second parallel flow path and is then brought together again. A flow-diverting means is provided at least in the first flow path in order to open or block the first flow path in a temperature-dependent manner.