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