In an elevator installation, an apparatus and a method use a support end connection for fastening a support device to an elevator car, a counterweight and/or a building. The support device has at least one cable or cable strand enclosed by a cable casing and is held in a wedge pocket by a wedge. The cable casing is formed of thermoplastic material or an elastomer and the cable or the cable strand is glued to, fused together with or mechanically connected with the cable casing in the region of the support end connection. A friction force transmitted from the support end connection to the cable casing can then be directly passed on to the load-bearing core of the support device, to the cables or to the cable strands. The tolerable tension force in the support device is increased. The support device is preferably a multiple cable.