Abstract: The invention is directed to a membrane fuel pump for an internal combustion engine equipped with a membrane carburetor. The engine is that of a working apparatus and especially a handheld portable tool such as a motor-driven chain saw or the like. The membrane pump includes a drive chamber charged with the pressure inside the crankcase of the engine and a pump chamber separated from the drive chamber by a membrane. The pump chamber is connected at its suction end to a fuel tank via a first check valve and is connected to a pressure controller of the membrane carburetor at its pressure end via a second check valve. The pressure end of the membrane pump is connected with the suction end thereof via a bypass and a throttle is mounted in this bypass. With the throttle bypass between the pressure end and the suction end, the pumped volume is reduced at idle and the pump pressure is smoothed. Disturbances at the membrane carburetor are avoided.