Machine for cutting paper logs, including a support for supporting one or more logs (3) disposed side by side, device for advancing the logs (3) on the support, device for the transverse cutting of the logs (3) and device for blocking logs (3) during their cutting. The cutting device includes an annular band blade (4) having a bevel (400, 401) on both its edges, provided in correspondence to a cutting station (T) in which the logs (3) are cut and connected to corresponding guiding and actuating structure, the blade (4) featuring a portion (40) defining a corresponding cutting plane which is oriented orthogonally with respect to the longitudinal axes of logs (3), the blade being provided with a sharpening device (100) continuously acting on both its bevels (400,401). The machine includes structure for the relative motion of logs (3) with respect to the blade (4) along the cutting plane. The blade (4) acts on one log (3) at a time.