We report the first measurement of the neutrino-oxygen neutral-current quasielastic (NCQE) cross section gamma It is obtained by observing nuclear deexcitation. rays which follow neutrino-oxygen interactions at the Super-Kamiokande water Cherenkov detector. We use T2K data corresponding to 3.01 x 10(20) protons on target. By selecting only events during the T2K beam window and with well-reconstructed vertices in the fiducial volume, the large background rate from natural radioactivity is dramatically reduced. We observe 43 events in the 4-30 MeV reconstructed energy window, compared with an expectation of 51.0, which includes an estimated 16.2 background events. The background is primarily nonquasielastic neutral-current interactions and has only 1.2 events from natural radioactivity. The flux-averaged NCQE cross section we measure is 1.55 x 10(-38) cm(2) with a 68% confidence interval of (1.22, 2.20) x 10(-38) cm(2) at a median neutrino energy of 630 MeV, compared with the theoretical prediction of 2.01 x 10(-38) cm(2).