A search for J/psi radiative decay to weakly interacting neutral final states was performed using the CLEO-c detector at the Cornell Electron Storage Ring. J/psi events were selected by observing the hadronic decay psi(2S) -> pi(+)pi(-) J/psi. A total of 3.7 x 10(6) J/psi events were used to study the decay J/psi -> gamma + X, where X is a narrow state that is invisible to the detector. No significant signal was observed, and upper limits on the branching fraction were set for masses m(X) up to 960 MeV/c(2). The upper limit corresponding to m(X) = 0 is 4.3 x 10(-6) at the 90% confidence level.