A machine for manufacturing tires (P) using at least one core (N) used as a reference surface of the manufacture of the tires (P) by moving consecutively from one manufacturing station to another from the start of assembly through to vulcanisation, comprising a first line of stations, oriented in a horizontal direction XX′, capable of receiving the core whose axis of rotation rr′ is oriented horizontally, and a second line of stations (200, 300, 300′), oriented in a horizontal direction YY′, perpendicular to the direction XX′, in which the stations are capable of receiving a core (N) whose axis rr′ is oriented vertically. The first line of stations comprises a first (120) and a second (130) station for assembling the tire including member (121, 131) for grasping and driving the core (N) in rotation about a horizontal axis rr′, merged with the axis XX′, such members being placed opposite one another. A turning device (110) is movable in the direction XX′ and is capable of transferring the core from one station to another within the first line of stations (120, 130) by making the axis rr′ of the core effect a translation along the axis XX′ and a rotation of 180° about a horizontal axis yy′ parallel to the direction YY′.