This machine includes a casing (2), a tool-holder spindle (4), a motor (6), a drive mechanism (12) having positive feed which mechanically connects the motor to the spindle and which includes a first drive member (32) for rotating the spindle, a second drive member (34) for driving the spindle in translation and which is screwed onto the spindle (4) in order to advance or return the spindle, and a clutch (36) having a return configuration which disengages the second member and the motor and an advance configuration which engages them. The clutch includes a rotary engagement member (44), a first key (48) which is movable between a position for connecting the rotary member to the motor and a position for disengaging the rotary member, and a second key (50) which is movable between a position for connecting the rotary member to the casing and a position for disengaging the rotary member.