A fuel supply systems with two series-connected pumps and with fuel injection valves that inject fuel directly into a combustion chamber of an engine in which a starting process takes less time because of the small feed quantity of the second fuel pump. The system includes a valve device which assures that during the starting process the first fuel pump furnishes the fuel to the fuel valves at elevated feed pressure. In many cases, this elevated feed pressure suffices to start the engine in the quickest possible time. The apparatus and the method are intended for an internal combustion engine of a vehicle.