Abstract: This machine comprises a train of grinding wheels (T) which is keyed on a shaft (2), and a blank-holding assembly (5, 7) designed to bring the blank (8) into contact with the grinding wheels. The said machine comprises, furthermore, an additional grinding wheel (9) mounted freely rotatably on a support (10), means (11) for actuating the support, which are designed to displace the additional grinding wheel (9) between a retracted position and an active position in which its axis of rotation (Y--Y) is parallel to the axis of the said shaft and in which the additional grinding wheel is in driving contact with a drive disc (17) integral with the train of grinding wheels, and means for bringing the blank into contact with the additional grinding wheel (9) when the latter is in the active position. The invention is used for the counter-bevelling and/or grooving and/or drilling of optical glasses.