A fuel supply system for a fuel cell stack includes a cathode gas supply line for supplying an oxidant to the cathode input of the fuel cell stack and an anode gas supply line for supplying fuel to the anode input of the fuel cell stack. An anode gas recirculation line recirculates partially used anode gas from the anode output to the anode input while a purge line may be connected for conducting away the partially used anode gas, as a purge gas from the anode gas recirculation line into the cathode gas supply line. The supply system also includes a conditioning arrangement disposed upstream of the cathode input. The conditioning arrangement is designed to a) mix a partial flow of the oxidant with the purge gas in a mixing station, b) compress accelerate, and/or heat the partial flow or the mixed flow, and c) conduct the mixed flow into the cathode gas supply line.