A tire includes at least one sidewall support exhibiting a substantially progressive increase in rigidity as the tire sags during depressurization. The support, which is disposed in the substantially median portion of the tire's sidewalls, includes a circumferential cord disposed in undulating manner in the sidewalls, whereby the axial position of the cord varies around the tire's circumference. At rated pressure, the structural rigidity of the sidewalls is that of a standard tire, while at reduced pressure the sidewalls rigidify in such a way as to be able to bear the load.