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We measure B(psi(2S)->gamma eta_c) = (4.32+/-0.16+/-0.60)x10^-3, B(J/psi->gamma eta_c)/B(psi(2S)->gamma eta_c) = 4.59+/-0.23+/-0.64, and B(J/psi->gamma eta_c) = (1.98+/-0.09+/-0.30)%. We observe a distortion in the eta_c line shape due to the photon-energy dependence of the magnetic dipole transition rate. 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