A method for mounting an endpiece for securing a tubular pipe and apparatus for implementing same. The method includes the steps of: a) providing an endpiece and a ring (36′) having an outer rotationally symmetrical cylindrical surface (40′), and a sheath having a sheath end (32) and an outer circular edge (46′); b) inserting the ring (36′) into the sheath end (32); c) leveling off the outer circular edge (46′); and d) covering the leveled edge with the endpiece. In step b) the ring (36, 36′, 36″) is inserted leaving one of the ends (44′) thereof extending out of the sheath end (32), and in step c) the outer circular edge (46′) is leveled off, by engaging the one of the two ring ends (44′).