A wire electrical discharge machine and a wire electrical discharge machining method are provided. The wire electrical discharge machine is equipped with a core position recognizing unit configured to recognize the position of a core which is fixed by a core fixing function, a core removal unit configured to remove the core, which is in the recognized position, from a workpiece on which machining has been performed, and a core removal determining unit configured to determine whether or not the core in the recognized position has actually been removed.