A self-aligning tap is used to produce correctly aligned new threads in an existing smooth bore hole. The design of the tap's shank, flutes, and cutting surfaces is that of commonly used existing taps. The flat end of the tap—the end that enters the existing bore hole—contains a threaded bore. A removable pilot that aligns the tap to the axis of the existing bore consists of a cylindrical body with a diameter slightly smaller than the existing bore hole, and a length no greater than the diameter of the major diameter of the tap. The upper end of the cylindrical pilot—the end that shoulders against the tap—comprises a threaded stem whose thread pitch matches that of the tap end's bore, and whose length is slightly less than the tap end bore's depth. Once the cylindrical pilot's stem is threaded into the tap end's bore and the resulting assembly is placed into the existing bore hole, it serves to align the tap so that its cutting teeth produce pilot threads which are accurately aligned to the existing bore hole's axis. Once the pilot threads have been produced by rotation of the tap, the tap may be removed from the existing bore hole, the pilot detached by unscrewing it, and the tap reinserted into the newly formed pilot threads. Further rotation of the tap completes the tapping operation of the existing bore hole to the desired depth.