The cross section for one-electron transfer H~ + He+ → H(1s) + He*(1s, n/m) is calculated in this paper. The collisional system is treated as a three-electron one and matrix elements for the formation of excited helium He* (1s, n/m) in both singlet and triplet states (1S, 2 1,3S, P and 3 1,3S, P, D) are obtained. Close-coupling calculations were done for the 27 〈1s, n/m | (1S, 2 1,3S, P and 3 1,3S, P, D) final states covering the range 1-10 × 107 cm s-1 of the relative collision velocity. An approximation is used for the effective potential of H- and Coulomb Green's functions are used to describe the weakly bound electron of H-. A satisfactory agreement is obtained with the experimental cross section.