We report high-precision calculations that correctly include the rearrangement channels by solving the modified Faddeev equations for energies between the Ps(n = 2) and H(n = 3) thresholds, which involve six and eight open channels. We find that 99% of the antihydrogen is formed in H(n = 2). Just above the Ps(n = 2) threshold the S, P, and D partial waves contribute nearly 1400pi(a(2)0) near the maximum. We find evidence that the induced long-range dipole potential is responsible for such a large H formation cross section. The possibility of utilizing this resonance to synthesize low-energy H is discussed.