The machine according to the invention comprises a store (3) containing the blanks (1); an extractor arm (13); means for shaping the blanks; and an upstream transport device (5) for receiving and distributing the blanks (6) of the boxes. The extractor arm (21) and the shaping means are provided on the frame (20) of a bench (4) which is inserted, as in a module, between the store (3) and the upstream transport device (5). Said bench (4) comprises: a ramp (24) which is upwardly inclined at an angle of 45 DEG in relation to the horizontal and acts as a guiding channel; a sole having an upper part, in the region of the shaping station (30), which can be retracted to enable the blank (6) to pass via the ramp (24); a transport system (32) for moving the blanks (1) from the lower point (B) to the upper point (H); a forming arm (40) carried by a framework (70) which extends above the ramp (24); and a transfer arm (41) which participates in the shaping of the blank (1) before moving it, once it has become a box blank (6), on the downstream transport device (5).