Detailed models for the UV burst from SN 1987A are calculated using a two-temperature radiation hydrodynamics code. The sequence of events which occur during shock breakout is described, including the formation of a radiative precursor, the disappearance of the shock, and the growth of a thin, dense shell above the photosphere which gives rise to a new viscous, isothermal shock. Continuum spectra are calculated under the assumption that the emergent radiation is a dilute blackbody with a color temperature equal to the radiation temperature at the thermalization depth. This color temperature is 2-3 times higher than the effective temperature. Light-travel-time corrections have been included in the light curves and the spectra. Quantities most relevant to accurate modeling of the fluorescing ring around SN 1987A are highlighted.