An adapter for a rolling assembly having an axis of rotation (YY′) includes an axially inner end connected to a rim, an axially outer end including an outer reinforcing element and intended to come into contact, via a substantially radial axially inner face, or bearing face, with a bead, a body connecting the axially outer end to the axially inner end so as to form a single piece, including at least one main reinforcement providing the connection between the outer reinforcing element and the inner reinforcing element, and including a substantially axial adapter seat intended to come into contact with a bead. The outer reinforcing element (15) is completely axially on the outside of the bearing face (21), and the outer reinforcing element (15) is a substantially annular structure, referred to as a bead wire, of substantially polygonal section, including at least two first layers of composite and/or metallic material, and at least two second layers of rubber compound, the first and second layers being arranged mutually parallel and in alternation with each other and parallel to the axis YY′.