The h(c)((1)P(1)) state of charmonium has been observed in the reaction psi(2S) --> pi(0)h(c) --> (gammagamma)(gammaeta(c)) using 3.08 x10(6) psi(2S) decays recorded in the CLEO detector. Data have been analyzed both for the inclusive reaction, where the decay products of the eta(c) are not identified, and for exclusive reactions, in which eta(c) decays are reconstructed in seven hadronic decay channels. We find M(h(c)) = 3524.4 +/- 0.6 +/- 0.4 MeV which corresponds to a hyperfine splitting DeltaM(hf)(1P) triple-bond <M((3)P(J)) - M((1)P(1)) = +1.0 +/- 0.6 +/- 0.4 MeV, and B(psi(2S) --> pi(0)h(c)) x B(h(c) --> gammaeta(c)) = (4.0 +/- 0.8 +/- 0.7) x 10(-4).