Abstract: To mount a multiple strand stay of a stayed bridge between a tower and a deck of said bridge, one of the strands (4.sub.1) that is to make up the stay is installed in its final oblique position; a guide cage is suspended from said strand installed in this way, the cage being constituted by a sequence of rigid hoops (5) that are interconnected by strings (6), said cage surrounding said first strand and also surrounding a traction cable (9); said cable is used to pull successive strands (4) up inside the cage; after each strand has been pulled up, it is put under tension so that it bears laterally against the already installed strands, and its ends are anchored respectively to the tower and to the deck; and after all of the strands have been installed, the guide cage is removed from the completed stay.