Abstract: The grinding apparatus permits precision machining of a valve seat or a sealing surface, and includes a grinding spindle having a holding and guiding part extending into a finished workpiece bore with clearance. The holding and guiding part is smaller in diameter than the grinding spindle and includes a pin which carries on its free end a grinding tool. The other end of the pin passes into the grinding spindle which has a substantially larger diameter. The grinding spindle is located outside the bore of the workpiece and may thus be made very stable. The pin is held by the grinding spindle so that the workpiece is held and guided securely even in the case of very small workpiece diameters.