  We report on a search for the radiative decay of Upsilon(1S) to the
pseudoscalar mesons eta and etaprime in 21.2 +/- 0.2 times 10^6 Upsilon(1S)
decays collected with the CLEO III detector at the Cornell Electron Storage
Ring (CESR). The eta meson was reconstructed in the three modes eta to
gamma-gamma, eta to pi+pi-pi0 and eta to 3pi0. The etaprime meson was
reconstructed in the mode etaprime to pi+ pi- eta with eta decaying through any
of the above three modes, and also etaprime to gamma rho, where rho decays to
pi^+ pi^-.
  Five out of the seven sub-modes are found to be virtually background-free. In
four of them we find no signal candidates and in one Upsilon(1S) to
gamma-etaprime, etaprime to pi+ pi- eta, eta to pi+pi-pi0 there are two good
signal candidates, which is insufficient evidence to claim a signal. The other
two sub-modes eta to gamma-gamma and etaprime to gamma rho are background
limited, and show no excess of events in their signal regions. We combine the
results from different channels and obtain upper limits at the 90% C.L. which
are B(Upsilon(1S) to gamma eta) < 1.0 times 10^-6 and B(Upsilon(1S) to gamma
etaprime) < 1.9 times 10^-6. Our limits are an order of magnitude tighter than
the previous ones and below the predictions made by some theoretical models.
