Fuel cell stack and method of operating a fuel cell system with such a fuel cell stack

A fuel cell stack comprising a plurality of fuel cells which are connected together. Each cell has an anode, a cathode, and a membrane therebetween and which are each arranged between two bipolar plates. The anode sides of the individual fuel cells have flow fields for a gaseous fuel supplied to the fuel cells extending between an anode space arranged at the top in the fuel cell stack and an anode space arranged at the bottom of the fuel cell stack. A collecting container is provided in the anode space arranged at the bottom of the fuel cell stack for water which is produced during the electrochemical reaction and is present in the form of a liquid and the water which is collected there has an evaporation surface and serves for the humidification of the fuel fed to the fuel cell stack.