Using a sample of (1310.6 +/- 7.0) x 10(6) J/psi events recorded with the Beijing Spectrometer III detector at the Beijing Electron Positron Collider II, we report the observation of the decay of the (1(1)S(0)) charmonium state eta(c) into a pair of omega mesons in the process J/psi -> gamma omega omega. The branching fraction is measured for the first time to be B(eta(c) -> omega omega) = (2.88 +/- 0.10 +/- 0.46 +/- 0.68) x 10(-3), where the first uncertainty is statistical, the second systematic, and the third is from the uncertainty of B(J/psi -> gamma eta(c)). The mass and width of the eta(c) are determined as M = (2985.9 +/- 0.7 +/- 2.1) MeV/c(2) and Gamma = (33.8 +/- 1.6 +/- 4.1) MeV.