An adapter, for a rolling assembly having an axis of rotation and comprising a tyre (P) having two beads (B), and a rim (J), provides the connection between one of the beads (B) and the rim (J), the said rim having two rim seats (7), the said adapter having an axially inner end (10) intended to be mounted on the rim seat (7) and comprising an inner reinforcing element (16), the said reinforcing element (16) having a centre C2, an axially outer end (9) comprising an outer reinforcing element (15), the said reinforcing element (15) having a centre C1, a body (11) that connects the said outer end (9) to the said inner end (10) so as to form a single piece and comprises at least one main reinforcement that provides the connection between the said outer reinforcer and the said inner reinforcer, and comprises a first end (17A) and a second end (17B), a substantially cylindrical adapter seat (18) intended to receive one of the said beads (B), the said seat (18) being situated at the axially outer end (9) of the said body (11), an adapter bearing face (21) substantially contained in a plane perpendicular to the axis of rotation, the said bearing face being situated on the axially inner face of the axially outer end (9), and an overall length L and a length LT measured between the centres C1 and C2. The first end (17A) and the second end (17B) of the reinforcement are arranged one upon the other so as to form an overlap.