An intervertebral-disc prosthesis for insertion into an intervertebral disc compartment has a first and a second constituent element (10,10′) each being provided on one side with a joint member (28,28′) and on the other side with an abutment face which abuts against adjacent vertebrae. The abutment faces have a convexly curved region that has at least substantially the shape of a ramp. Preferably the region is surrounded by an annular flat region having a larger roughness than the curved region. The curved region may have a vertex which contacts a vertex of a dome formed in the adjacent vertebra. This facilitates rotation of the constituent element within the intervertebral disc compartment.