A cutting-off machine for the transversal cutting of logs of paper material, including a structure on which the logs to be cut are moved and a cutting unit including a support plate for a blade arranged along a lying plane. The machine includes at least one sharpening unit with two grinding wheels for sharpening the blade on opposite sides of the plane and equipped with an abrasive side and a grinding wheel positioning device with respect to the blade, arranged on each sharpening unit, by which each grinding wheel is placed in a position of contact with the blade in a phase of sharpening the latter starting from a position initial inoperative. The positioning device is equipped with automatic elements for detecting the wear of the abrasive side of the grinding wheels.