Total intervertebral joint prosthesis, comprising: - two pairs (2, 3) of elongated joint elements designed to be placed one above the other, one element (6, 7) of a pair of joint elements comprising a rounded disc joint surface (6e, 7e), a facet joint surface (6b, 7b), and a connecting arm (6c, 7c) extending between that disc joint surface (6e, 7e) and facet joint surface (6b, 7b), and the other element (8, 9) of that pair of joint elements comprising a rounded disc joint surface (8e, 9e), a facet joint surface (8b, 9b), and a connecting arm (8c, 9c) extending between that disc joint surface (8e, 9e) and facet joint surface (8b, 9b); when the two elements (6, 8; 7, 9) of a same pair (2, 3) are placed one above the other, the rounded disc joint surface (6e, 7e) of one element (6, 7) cooperates with the rounded disc joint surface (8e, 9e) of the other element (8, 9), and the facet joint surfaces (6b-9b) of the two elements cooperate with one another; - means (10a, 10b, 11 a, 11 b) for positioning one pair (2, 3) of elements relative to the other; wherein - each disc joint surface (6e-9e) is in the shape of a hemisphere portion, that hemisphere having a same radius from one pair (2, 3) of joint elements to the next; - said positioning means comprise at least one first posterior positioning surface (10a, 11 a) arranged on at least one element (6, 7) of a first pair of joint element (6, 7)s and at least one second posterior positioning surface (10b, 11 b) arranged on at least one element (8, 9) of the second pair of joint elements; characterised in that - the prosthesis is associated with at least one rigid transverse element (4, 5; 21) that is capable to be connected to said first posterior positioning surface (10a, 11 a) and said second posterior positioning surface (10b, 11 b), this rigid transverse element (4, 5) comprising at least one third positioning surface (12a, 13a; 22a) designed to be placed against said first posterior positioning surface (10a, 11a) and at least one fourth positioning surface (12b, 13b; 22b) designed to be placed against said second posterior positioning surface (10b, 11 b), this placement of said third and fourth positioning surfaces (12a à 13b; 22a, 22b) against said first and second positioning surfaces (10a-11b), respectively, making it possible to perform relative positioning of one pair of joint elements (6, 8) relative to the other pair of joint elements (7, 9) such that the disc joint surfaces (6e, 7e, 8e, 9e) of the joint elements are situated in a same geometric hemisphere. - two pairs (2, 3) of elongated joint elements designed to be placed one above the other, one element (6, 7) of a pair of joint elements comprising a rounded disc joint surface (6e, 7e), a facet joint surface (6b, 7b), and a connecting arm (6c, 7c) extending between that disc joint surface (6e, 7e) and facet joint surface (6b, 7b), and the other element (8, 9) of that pair of joint elements comprising a rounded disc joint surface (8e, 9e), a facet joint surface (8b, 9b), and a connecting arm (8c, 9c) extending between that disc joint surface (8e, 9e) and facet joint surface (8b, 9b); when the two elements (6, 8; 7, 9) of a same pair (2, 3) are placed one above the other, the rounded disc joint surface (6e, 7e) of one element (6, 7) cooperates with the rounded disc joint surface (8e, 9e) of the other element (8, 9), and the facet joint surfaces (6b-9b) of the two elements cooperate with one another; - means (10a, 10b, 11 a, 11 b) for positioning one pair (2, 3) of elements relative to the other; wherein - each disc joint surface (6e-9e) is in the shape of a hemisphere portion, that hemisphere having a same radius from one pair (2, 3) of joint elements to the next; - said positioning means comprise at least one first posterior positioning surface (10a, 11 a) arranged on at least one element (6, 7) of a first pair of joint element (6, 7)s and at least one second posterior positioning surface (10b, 11 b) arranged on at least one element (8, 9) of the second pair of joint elements; characterised in that - the prosthesis is associated with at least one rigid transverse element (4, 5; 21) that is capable to be connected to said first posterior positioning surface (10a, 11 a) and said second posterior positioning surface (10b, 11 b), this rigid transverse element (4, 5) comprising at least one third positioning surface (12a, 13a; 22a) designed to be placed against said first posterior positioning surface (10a, 11a) and at least one fourth positioning surface (12b, 13b; 22b) designed to be placed against said second posterior positioning surface (10b, 11 b), this placement of said third and fourth positioning surfaces (12a à 13b; 22a, 22b) against said first and second positioning surfaces (10a-11b), respectively, making it possible to perform relative positioning of one pair of joint elements (6, 8) relative to the other pair of joint elements (7, 9) such that the disc joint surfaces (6e, 7e, 8e, 9e) of the joint elements are situated in a same geometric hemisphere.