We report the first discovery of a QSO damped Ly-alpha (DLA) system by the GALEX satellite. The system was initially identified as an MgII absorption-line system (z_abs=1.028) in the spectrum of SDSS QSO J0203-0910 (z_em=1.58). The presence of unusually strong absorption due to metal lines of ZnII, CrII, MnII, and FeII clearly suggested that it might be a DLA system with N{HI} > 2 x 10^20 atoms cm^-2. Follow-up GALEX NUV grism spectroscopy confirms the system exhibits a DLA absorption line, with a measured HI column density of N{HI} = 1.50+/-0.45 x 10^21 atoms cm^-2. By combining the GALEX N{HI} determination with the SDSS spectrum measurements of unsaturated metal-line absorption due to ZnII, which is generally not depleted onto grains, we find that the system's neutral-gas-phase metal abundance is [Zn/H] = -0.69+/-0.22, or ~20% solar. By way of comparison, although this system has one of the largest Zn^+ column densities, its metal abundances are comparable to other DLAs at z~1. Measurements of the abundances of Cr, Fe, and Mn help to further pin down the evolutionary state of the absorber.