The machine includes a tire-building zone for producing, on a drum, a carcass entity that includes a carcass ply with carcass reinforcers. The machine also includes a finishing zone for producing a crown entity that includes a tread, a shaping system for causing the carcass entity to expand radially in order to give it a toroidal shape. In an assembling zone, the crown entity is assembled on the carcass entity. A biasing system interacts, during the shaping, with the carcass reinforcers so as to modify the angular orientation of the carcass reinforcers with respect to the circumferential direction of the drum. An automatic control unit allows a biased-crown tire to be manufactured in this way.