  We report evidence for the ground state of bottomonium, eta_b(1S), in the
radiative decay Upsilon(3S) --> gamma eta_b in e^+e^- annihilation data taken
with the CLEO III detector. Using 6 million Upsilon(3S) decays, and assuming
Gamma(eta_b) = 10 MeV/c^2, we obtain B(Upsilon(3S) --> gamma eta_b) = (7.1 +-
1.8 +- 1.1) X 10^{-4}, where the first error is statistical and the second is
systematic. The statistical significance is about 4 sigma. The mass is
determined to be M(eta_b) = 9391.8 +- 6.6 +- 2.0 MeV/c^2, which corresponds to
the hyperfine splitting Delta M_{hf}(1S)_b = 68.5 +- 6.6 +- 2.0 MeV/c^2. Using
9 million Upsilon(2S) decays, we place an upper limit on the corresponding
Y(2S) decay, B(Y(2S) --> gamma eta_b) < 8.4 X 10^{-4} at 90 % confidence level.
