We report on a search for the radiative decay Y(1S) --> gamma eta' in 61.3
pb(-1) of data taken with the CLEO II detector at the Cornell Electron Stor
age Ring. Three decay chains were investigated, all involving eta' --> pi (
+) pi (-) eta, followed by eta --> gamma gamma, eta --> pi (0)pi (0)pi (0),
or eta --> pi (+) pi (-) pi (0). 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., si
gnificantly smaller than the previous limit. We compare our result to other
radiative Y decays, to radiative J/psi decays, and to theoretical predicti
ons.