  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 2.92 M_sun and 0.19 M_sun
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 kK 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~hours. 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.
