A vertebral disc prosthesis, notably for cervical vertebrae, comprising two components (2, 3) intended to be connected to the respective vertebral plates which are jointed to each other, and at least one component in an elastically deformable material; a first of these two components comprises a bottom, a wall and, at a distance from said bottom, an edge extending inwards; the second component comprises a plate intended to come into contact with the vertebral plate of the relevant vertebra and a pin with a widened head; - said wall of said first component is a peripheral wall which delimit with said bottom a single central housing, said bottom forming a first articular surface, and said edge forming, on the side opposite to said bottom, a peripheral supporting surface and, on the side turned towards said bottom, a second articular surface; - said pin of said second component is a single central pin, and said widened head forms, at its free end, a third articular surface able to cooperate with said first articular surface, and, on its side turned to the side of said plate, a fourth articular surface able to cooperate with said second articular surface; - said housing and said head are mutually dimensioned so that both components (2, 3) are axially mobile relatively to each other, i.e. along the axis of this central pin; - said component in an elastically deformable material is interposed between said peripheral supporting surface of the first component and said plate of the second component, this component in an elastically deformable material normally maintaining said second and fourth articular surfaces (13, 24) in contact with each other, and being able to be compressed so that said first and third articular surfaces (11, 23) come into contact with each other; characterized in that said housing and said head are mutually dimensioned so that both components (2, 3) are mobile relatively to each other, sideways, i.e. perpendicularly to the axis of said central pin; the possible sideways displacement of first and second components relatively to each other is such that: - in the case of an axially exerted force, but on one side of the prosthesis, the deformable component is only compressed on this side; it thus allows said first and third articular surfaces to come into contact on this same side but, on the opposite side, it maintains the contact of said second and fourth articular surfaces; - in the case of a force tending to displace an component in translation relatively to the other component in a transverse direction relatively to the vertebrae, the deformable component allows the second and fourth articular surfaces to slide on each other on the side opposite to the one on which said force is exerted, while allowing contact of the first and third articular surfaces on the side on which said force is exerted. - said wall of said first component is a peripheral wall which delimit with said bottom a single central housing, said bottom forming a first articular surface, and said edge forming, on the side opposite to said bottom, a peripheral supporting surface and, on the side turned towards said bottom, a second articular surface; - said pin of said second component is a single central pin, and said widened head forms, at its free end, a third articular surface able to cooperate with said first articular surface, and, on its side turned to the side of said plate, a fourth articular surface able to cooperate with said second articular surface; - said housing and said head are mutually dimensioned so that both components (2, 3) are axially mobile relatively to each other, i.e. along the axis of this central pin; - said component in an elastically deformable material is interposed between said peripheral supporting surface of the first component and said plate of the second component, this component in an elastically deformable material normally maintaining said second and fourth articular surfaces (13, 24) in contact with each other, and being able to be compressed so that said first and third articular surfaces (11, 23) come into contact with each other; - in the case of an axially exerted force, but on one side of the prosthesis, the deformable component is only compressed on this side; it thus allows said first and third articular surfaces to come into contact on this same side but, on the opposite side, it maintains the contact of said second and fourth articular surfaces; - in the case of a force tending to displace an component in translation relatively to the other component in a transverse direction relatively to the vertebrae, the deformable component allows the second and fourth articular surfaces to slide on each other on the side opposite to the one on which said force is exerted, while allowing contact of the first and third articular surfaces on the side on which said force is exerted.