Vertebral osteosynthesis equipment including at least one connecting bar, at least one flexible ligament, and at least one connector making it possible to connect said ligament to said connecting bar, the connector including a first conduit for engaging the connecting bar, equipped with a first tapped bore and a first screw for immobilizing the connector relative to said connecting bar, and a second conduit for receiving at least one strand of the ligament, having an insertion opening for this or these strands and an outlet opening for this or these strands; the connector includes a first portion in which said first conduit is formed and a second portion in which the second conduit is formed, said second conduit being separate from the first conduit and not in communication therewith; - said second conduit is equipped with means (19, 20) for immobilizing the ligament relative to the connector, including a tightening member exerting, when it is tightened, a gripping of the ligament in said second conduit along an axis perpendicular to said second conduit; - said second conduit is rectilinear between said insertion opening and said outlet opening, and said second conduit has a transverse section adjusted to the transverse section of the strand(s) of the ligament that it is intended to receive, i.e. a transverse section just sufficient to allow sliding of the strand(s) of the ligament in said second conduit, but such that the strand(s) are in close contact with the walls delimiting said second conduit; characterized in that - said means (19, 20) for immobilizing the ligament relative to the connector include a second tapped bore formed in said second portion, with an axis perpendicular to said second conduit and communicating with that conduit, and said tightening member is a second screw capable of being received in said second tapped bore, having a planar and smooth inner end surface; - said second screw has a diameter larger than the width of the strand of the ligament engaged in said second conduit, or, when the two strands of the ligaments are engaged in said conduit side-by-side, has a diameter larger than the cumulative width of said two strands. - said second conduit is equipped with means (19, 20) for immobilizing the ligament relative to the connector, including a tightening member exerting, when it is tightened, a gripping of the ligament in said second conduit along an axis perpendicular to said second conduit; - said second conduit is rectilinear between said insertion opening and said outlet opening, and - said means (19, 20) for immobilizing the ligament relative to the connector include a second tapped bore formed in said second portion, with an axis perpendicular to said second conduit and communicating with that conduit, and said tightening member is a second screw capable of being received in said second tapped bore, having a planar and smooth inner end surface; - said second screw has a diameter larger than the width of the strand of the ligament engaged in said second conduit, or, when the two strands of the ligaments are engaged in said conduit side-by-side, has a diameter larger than the cumulative width of said two strands.