This device comprises a connection element (4) between a rod (1), or other longitudinal implant, and a bone anchorage screw (3) in the degenerative vertebra. This connection element (4) includes a ring (8) so dimensioned that the rod (1) is capable of extending therethrough. The ring is provided with screws (14) for clamping to the rod (1) and is radially extended by a cylindrical arm (9) adapted to be secured to the bone anchorage screw (3) and to be clamped on the screw. The arm (9) and the ring (8) constitute a unit in one piece. The invention enables the surgeon to avoid having to produce additional deformations of the rod (1) when it is in presence of non-aligned pedicles by leaving to the surgeon complete liberty as to the position of the two axes of the bone anchorage screw (3) and of the rod (1). The invention further contemplates using the lateral connection element to facilitate contouring the longitudinal implant within the patient, or in situ to segmentally and globally correct spinal deformities in all degrees of freedom.