Let G be a finite primitive permutation group of degree n and rank three. It is readily seen that one of the following holds: (i) \(T\times T\triangleleft G\leq T_ 0 wr Z_ 2\), where \(T_ 0\) is a 2-transitive group of degree \(n_ 0\), the socle T of \(T_ 0\) is simple and \(n=n^ 2_ 0\); (ii) G is an affine group; (iii) the socle L of G is simple. The aim of the present paper is to complete the list of groups G satisfying (iii), using the recent classification of finite simple groups. The paper settles the cases L an exceptional group of Lie type or a sporadic group; the remaining cases being dealt with in previous publications by \textit{E. Bannai} [J. Fac. Sci., Univ. Tokyo, Sect. I A 18, 475-486 (1972; Zbl 0262.20003)] and \textit{W. M. Kantor} and \textit{R. A. Liebler} [Trans. Am. Math. Soc. 271, 1-71 (1982; Zbl 0514.20033)]. The information about G in case (i) can be drawn from the classification of 2-transitive permutation groups and in case (ii) from a preprint by the first author.