Abstract: The invention relates to an intersomatic implant designed to be inserted in the disk space defined between two adjacent vertebrae, namely an overlying vertebra and an underlying vertebra, for the purpose of reestablishing the anatomic space between the vertebrae, the implant being in the form of a cage (1) that is generally in the shape of a rectangular block having at least two sagittal walls (2, 3) interconnected at least by an anterior transverse wall (4) and by a posterior transverse wall (5), the walls (2 to 5) presenting rims (10) extending on one surface to define a first transverse face (8) and on the other side to define a second transverse face (9). According to the invention, the implant comprises: a first transverse face (8) presenting in the sagittal plane a convex profile congruent with the sagittal anatomic profile of an overlying vertebra; and a second transverse face presenting in the frontal plane a convex profile congruent with the frontal anatomic profile of an overlying vertebra.