In a fuel injector for injecting fuel into a combustion chamber of an internal combustion engine, an injection valve member for opening and closing at least one injection opening is triggered by a control valve embodied as a magnet valve. On the armature of the magnet valve, a sealing face is embodied that for closing the control valve is positionable into a valve seat. The armature of the magnet valve is movable, without armature guidance, between an upper and a lower stroke stop. Because the sealing face, which for closing the control valve is positionable into the valve seat, is embodied on the armature, an additional closing element of the kind provided in the prior art can be dispensed with.