The perturbation series for the spin-flip amplitude, where two units of spin are exchanged, of the scattering of two vector mesons through fermion-fermion-vector-meson coupling is studied in the limit of high energies. Similar to the case of elastic scattering (without spin flip) in quantum electrodynamics, there are, for this spin-flip amplitude, terms in the perturbation series of the orders of magnitudes s, s lns, s (lns)2, s (lns)3, etc. For each n, the leading term in the coefficient of s (lns)n is due to Feynman diagrams with n+2 closed fermion loops. These leading terms are found explicitly for the exactly forward direction, and are then summed. Because these terms alternate in sign, at high energies this sum is smaller than any one of the leading terms in the series. This sum is studied in detail for the case where massless photons are exchanged, and is found to have a simple factorization property. Some of the results are extended to the more general case of exchanging massive neutral vector mesons. The high-energy behavior of this spin-flip amplitude is qualitatively different from that of the spin-nonflip amplitude studied previously.