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The binary star BE Ursae Majoris is recently emerged from the common envelope phase; indeed, the hot sdO/DAO component is the central star of the associated planetary nebula. As such, BE UMa represents an important test case of stellar evolution theory. Using the Hubble Space Telescope (HST) Goddard High Resolution Spectrograph (GHRS), we measured the radial velocity amplitude of the He II λ1640 absorption line from the sdO/DAO component of this eclipsing system. Combining our results with those of Crampton, Cowley, & Hutchings, we determine stellar masses in units of solar mass as follows: for the sdO, the mass is 0.70 ± 0.07, and that of the secondary star is 0.36 ± 0.07, where we report the 1 σ value for all errors. The separation between the component stars is 7.5 R☉ ± 0.5 R☉ and is insensitive to small changes in inclination angle due to the near edge-on viewing angle of 84° ± 1°. Using these values, we modeled the eclipse light curve. Our results matched observed UBVR light curves of Wood and coworkers only if the modeled secondary star radius of 0.72 R☉ ± 0.05 R☉ has nearly double the radius expected from the main-sequence mass-radius relation. The secondary star has thus not yet relaxed to thermal equilibrium since the common envelope phase ended ~104 yr ago. Using the λ1640 absorption-line profile and the surrounding continuum, we also were able to constrain the sdO helium abundance as log nHe=-1.1 ± 0.2 and log nFe < 1. Our results support the sdO/DAO log g ~6.5 surface gravity and Teff~100,000 K values of Liebert et al. and are consistent with the post-AGB evolutionary track. Our best estimate of the distance to the BE UMa system is 2000 pc.