The invention relates to a fuel injector, in particular a common rail injector, for injecting fuel into a combustion chamber of an internal combustion engine. The fuel injector has an injection valve element which is adjustable between a closing position and an opening position and which is switchable by means of a control valve. The control valve has a sleevelike control valve element that is adjustable along an adjustment axis and in its closing position rests sealingly on a control valve seat element. According to the invention, it is provided that the control valve seat element is disposed movably relative to the adjustment axis of the control valve element.