  We report on a search for the radiative decay U(1S) -> gamma + eta' in 61.3
pb^-1 of data taken with the CLEO II detector at the Cornell Electron Storage
Ring. Three decay chains were investigated, all involving eta' -> pi+ pi- +
eta, followed by eta -> gamma + gamma, eta -> pi0 + pi0 + pi0, or eta -> pi+ +
pi- + pi0. We find no candidate events in any of the three cases and set a
combined upper limit of 1.6 x 10^-5 at 90% C.L., significantly smaller than the
previous limit. We compare our result to other radiative U(1S) decays, to
radiative J/psi decays, and to theoretical predictions.
