A tire with radial carcass reinforcement, wound in each bead B on at least one bead wire (2) to form an upturn (10), the radially inner face of each said bead being substantially dimensionally identical to the face of the seat of its intended mounting rim, characterized in that, when mounted on the rim and inflated to its operating pressure, its carcass reinforcement, viewed in meridian section, is wound in at least one of the beads B about a bead wire (2) coated with rubber mix, passing from the heel to the toe of said at least one bead B, the upturn (10) extending in a profile (3) made of rubber mix in the form of a wedge defined by two sides (31 and 32) coming from an apex A located beneath the section of the coated bead wire (2), the radially outer side (31) forming, with a line parallel to the axis of rotation passing through said apex A, an acute angle .phi..sub.1 open radially towards the outside, and the radially inner side (32) forming with said parallel line an acute angle .phi..sub.2, open radially towards the inside, the rubber mix forming the wedge profile (3), axially adjacent to the coated bead wire (2), having a Shore A hardness of at least 65 and greater than the Shore A hardness of any rubber mixes located axially or radially outside the coated bead wire (2) and the wedge profile (3).