In a direct-methanol fuel cell stack with an at least partial circulation (7, 9) of the fuel, according to the invention a means is arranged in this circulation for reducing the concentration of an undesired byproduct. This means is comprised especially of a further electrochemical cell (8) to which electric current is applied and which at least partly transforms the byproducts by an electrochemical reduction reaction again to methanol and water. Since the concentration of byproducts is small, the loss in electric current required for the additional reduction reaction has no noticeable effect on the efficiency of the fuel cell stack.