A cervical disc prosthesis or intervertebral disc prosthesis that includes a first disc comprising a protrusion connected to a disc body and a convex surface facing a first cervical vertebra when implanted, and a second disc comprising a socket recessed in a disc body, and a convex surface facing a second cervical vertebra when implanted. The second disc remains substantially parallel to the first disc and axially spaced from the first disc. The protrusion is adapted to fit into the socket to join the first disc with the second disc, and the protrusion is fit for pivoting movement into the socket. Also provided is a method for implanting the inventive cervical disc prosthesis or the inventive intervertebral disc prosthesis into an intervertebral disc area defined between two adjacent vertebrae.