A fuel cell system is described with a so-called direct methanol fuel cell (DMFC). To minimize the methanol crossover the concentration of the methanol/water mixture supplied to the anode of the fuel cell is made dependent on the magnitude of the electrical power takeoff. In order to be able to adapt the concentration of the methanol quickly in the case of a load change, it is proposed that a separating column be introduced into the anode cycle which reduces the concentration of the methanol. To achieve adapted increases in the concentration of methanol, both a concentrated methanol/water mixture from a secondary cycle and possibly additionally pure methanol from a tank are supplied. In the case of a reduction in the power demand only the secondary cycle needs to be blocked with the result that a lean mixture is available immediately.