A fuel cell system having a fuel cell with an anode compartment and a cathode compartment. The fuel cell system is also provided with a partial-pressure pressure swing adsorption assembly operating at low pressure which uses supply gas for the cathode compartment of the fuel cell to separate carbon dioxide from the anode exhaust gas exiting from the anode compartment of the fuel cell. The partial-pressure pressure swing adsorption assembly is adapted to carry out this separation based on the difference in partial pressures of the carbon dioxide in the anode exhaust gas and the oxidant gas in the supply gas. The fuel cell system utilizes a further unit to recycle the exhaust gas, after separation of the carbon dioxide, to the anode compartment and to feed the supply gas with the carbon dioxide to the cathode compartment.