We report the serendipitous detection by GALEX of fast (<1 day) rising (>1 mag) UV emission from two Type II plateau (II-P) supernovae (SNe) at z=0.185 and 0.324 discovered by the Supernova Legacy Survey. Optical photometry and VLT spectroscopy 2 weeks after the GALEX detections link the onset of UV emission to the time of shock breakout. Using radiation hydrodynamics and non-LTE radiative transfer simulations, and starting from a standard red supergiant (RSG; Type II-P SN progenitor) star evolved self-consistently from the main sequence to iron core collapse, we model the shock breakout phase and the 55 hr that follow. The small scale height of our RSG atmosphere model suggests that the breakout signature is a thermal soft X-ray burst (lambda_peak ~ 90AA) with a duration of <~ 2000 s. Longer durations are possible but require either an extended and tenuous non-standard envelope, or an unusually dense RSG wind with dot{M} ~ 10^(-3) Msun yr^(-1). The GALEX observations miss the peak of the luminous (M_FUV ~ -20) UV burst but unambiguously capture the rise of the emission and a subsequent 2 day long plateau. The postbreakout, UV-bright plateau is a prediction of our model in which the shift of the peak of the spectral energy distribution (SED) from ~100 to ~1000AA and the ejecta expansion both counteract the decrease in bolometric luminosity from ~10^11 to ~10^9 L_sun over that period. Based on the observed detection efficiency of our study we make predictions for the breakout detection rate of the GALEX Time Domain Survey.
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