Abstract: In the process or internally grinding bores with the aid of a slightly conical grinding wheel (15) in a computer controlled manner, the rotational axis (14) of the grinding wheel is adjusted to a small angle (.alpha.) in relation to the rotational axis (3a) of the workpiece (5). The grinding wheel is sharpened along a generatrix which lies opposite the generatrix of the wheel when the wheel is in rough grinding engagement with the bore. The rough grinding operation is carried out at a feed pressure such that twice this angle (i.e. 2(.alpha.) is levelled-out or eliminated and the respective generatrices of the wheel and the bore will substantially coincide. The fine grinding phase of the grinding operation is carried out with substantially coinciding bore and grinding wheel generatrices along an opposite bore generatrix in relation to the rough grinding engagement location. The invention also relates to a grinding machine for carrying out the method.