KOI-81 is a totally eclipsing binary discovered by the Kepler mission that consists of a rapidly rotating B-type star and a small, hot companion. The system was forged through large-scale mass transfer that stripped the mass donor of its envelope and spun up the mass gainer star. We present an analysis of UV spectra of KOI-81 that were obtained with the Cosmic Origins Spectrograph on the Hubble Space Telescope that reveal for the first time the spectral features of the faint, hot companion. We present a double-lined spectroscopic orbit for the system that yields mass estimates of and for the B-star and hot subdwarf, respectively. We used a Doppler tomography algorithm to reconstruct the UV spectra of the components, and a comparison of the reconstructed and model spectra yields effective temperatures of 12 and 19?27 kK for the B-star and hot companion, respectively. The B-star is pulsating, and we identified a number of peaks in the Fourier transform of the light curve, including one that may indicate an equatorial rotation period of 11.5 hr. The B-star has an equatorial velocity that is 74% of the critical velocity where centrifugal and gravitational accelerations balance at the equator, and we fit the transit light curve by calculating a rotationally distorted model for the photosphere of the B-star.