For finishing round surface sections of very heavy workpieces, for example the main pin and crank pin of crankshafts (12) for marine diesel engines, the invention proposes that the finishing belt and the pressure bodies (76) over which it passes be mounted on a rotary support ring (52) driven by a motor (74), while the workpiece remains stationary in the direction of rotation during the finish machining. The support ring (52) and a rotationally fixed head frame (44) of the finish machining head (34) are each split along a vertical contact plane running through the axis of the support ring, so that one half (42) of the machining head can be moved away from the other half (40) to allow the workpiece section for machining to be inserted into and removed again from the machining head.