The invention relates to a tyre with a radial carcass reinforcement (2) made up of at least one layer of reinforcing elements which is anchored in each of the beads (3) by being wrapped around a bead wire (4), said wrapped-around portion (7) of carcass reinforcement being reinforced with at least one layer of reinforcing or stiffening elements (8), the radially outer end of which is radially on the outside of the end (9) of the wrapped-around portion (7). According to the invention, the reinforcing elements of at least one stiffener are unwrapped metal cords with saturated layers which on the test known as the permeability test return a flow rate of less than 5 cm3/min and the elastic modulus of the layer (11) of polymer compound in contact with the wrapped-round portion of carcass reinforcement and which separates the stiffener from the end of the wrapped-around portion of carcass reinforcement is higher than 4 MPa.