The proposed sheath is for a structural cable (10) having a path between an upper anchorage (16) and a lower anchorage (17). It comprises sheath segments (21) assembled along the path of the structural cable, at least one supporting rope (30) extending along the sheath segments and having an upper end connected to the construction work adjacent to the upper anchorage, and connectors (32) for connecting the sheath segments to the at least one supporting rope. The connectors (32) are configured to block relative upward movement of the supporting rope (30) with respect to the sheath segments (21) and to allow relative downward movement of the supporting rope with respect to the sheath segments.
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