A device for braking an elongated rotating body, includes at least two parts with coaxial conical surfaces, able to be slipped onto the body, and one, a so-called male cone, mounted integrally in rotation with the body, and the other, a so-called female cone, mounted fixed in rotation relative to the body. The female cone assumes the shape of a wheel with an axial bore with a conical bearing, and the male cone has a center recess. The cones are controlled in terms of axial relative movement by a control element. The male cone is, when the cones are close to one another, corresponding to the braking position, housed inside the axial bore of the female cone, with its conical external peripheral surface being held, by contact with a support, with the conical bearing of the female cone.