  We have searched for the bottomonium eta_b(1S) via the hindered magnetic
dipole (M1) photon transition Upsilon(3S) -> gamma eta_b(1S). No evidence for
such a transition is found in the data sample of 4.7 x 10^6 Upsilon(3S)'s
collected with the CLEO-III detector. We set upper limits on the branching
ratio from 9.30 to 9.43 GeV/c^2 of eta_b(1S) masses. These upper limits rule
out many previously published phenomenological estimates of the rate for this
transition.
