We report the first discovery of a QSO damped Lyα system (DLA) by the Galaxy Evolution Explorer (GALEX) satellite. The system was initially identified as an Mg II absorption-line system (z _abs = 1.028) in the spectrum of Sloan Digital Sky Survey (SDSS) QSO J0203-0910 (z _em = 1.58). The presence of unusually strong absorption due to metal lines of Zn II, Cr II, Mn II, and Fe II clearly suggested that it might be a DLA with N _{H I} ≥ 2 × 10²⁰ atoms cm^–2. Follow-up GALEX NUV grism spectroscopy confirms that the system exhibits a DLA absorption line, with a measured H I column density of N _{H I} = 1.50 ± 0.45 × 10²¹ atoms cm^–2. By combining the GALEX N _{H I} determination with the SDSS spectrum measurements of unsaturated metal-line absorption due to Zn II, which is generally not depleted onto grains, we find that the system's neutral-gas-phase metal abundance is [Zn/H] = –0.70 ± 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.