A machine for machining a spiral cutting edge on an elongate workpiece having a rounded head by moving the workpiece relative to a rotating tool comprises a support unit mounted for vertical translational movement and rotation about a vertical axis. The support unit carries a headstock which can be positioned to make the axis of its workpiece-carrying spindle intersect the vertical axis, and a device for helicoidally driving the headstock spindle. The support unit rests on an inclined ramp surrounding the vertical axis and along which the unit can roll to helicoidally move the headstock for spirally machining a cutting edge into the rounded end of the workpiece. Thereafter a spiral cutting edge is machined into the cylindrical shaft of the workpiece, smoothly continuing the rounded, spiral edge.