A cutting-off machine, comprising a cutting unit (2) being enabled to receive a tube (100) and to cut the tube (100) according to a plane transversal to a longitudinal axis (X) of the tube (100), movement means (8) enabled to grip stably a length (S) of cut tube (100) and move said length (S) away from the cutting unit (2) along a direction of advance (A) of the tube (100), and a cutting-off device (5) interposed between said cutting unit (2) and said movement means (8) and configured to realise cutting off of an inner welding seam (200) floating inside said tube (100).