We report on a search for the radiative decay Upsilon(1S)-->gammaeta(') 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-->gammagamma, 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 1.6x10(-5) at 90% C.L., significantly smaller than the previous limit. We compare our result to other radiative Upsilon decays, to radiative J/psi decays, and to theoretical predictions.