This hot-marking machine (1) comprises a marking member (6) intended to press a tape (7) against the peripheral surface (2A) of individual parts (2) in order to mark them, said individual parts (2) being positioned in succession in a marking station and being marked by the repetition of a marking sequence for each part. The machine (1) comprises means (12-14) for unwinding the tape, intended to unwind a length of tape (7) sequentially at the end of each marking sequence, for the next marking sequence, and means (15-17) for rewinding the tape. The machine (1) further includes a tape suction box (18) and a tape braking surface (19A), the braking surface (19A) being of fixed orientation relative to the frame (3) of the machine (1) during each marking sequence, the tape (7) output by the unwinding means (12-14) passing in succession through the suction box (18), along the braking surface (19A) and facing the marking member (6), in such a way that a reserve of tape (7) is available in the suction box (18) between the unwinding means (12-14) and the braking surface (19A), while the tape (7) is tensioned, facing the marking member (6), between the braking surface (19A) and the rewinding means (15-17).