Usingthe “Scenario Machine” (a specialized numerical code formodeling the evolution of large ensembles of binary systems), we have studied the physical properties of rapidly rotating main-sequence binary stars (Be stars) with white-dwarf companions and their abundance in the Galaxy. The calculations are the first to take into account the cooling of the compact object and the effect of synchronization of the rotation on the evolution of Be stars in close binaries. The synchronization time scale can be shorter than the main-sequence lifetime of a Be star formed during the first mass transfer. This strongly influences the distribution of orbital periods for binary Be stars. In particular, it can explain the observed deficit of short-period Be binaries. According to our computations, the number of binary systems in the Galaxy containing a Be star and white dwarf is large: 70–80% of all Be stars in binaries should have degenerate dwarf companions. Based on our calculations, we conclude that the compact components in these systems have high surface temperatures. Despite their high surface temperatures, the detection of white dwarfs in such systems is hampered by the fact that the entire orbit of the white dwarf is embedded in the dense circumstellar envelope of the primary, and all the extreme-UV and soft X-ray emission of the compact object is absorbed by the Be star’s envelope. It may be possible to detect the white dwarfs via observations of helium emission lines of Be stars of not very early spectral types. The ultraviolet continuum energies of these stars are not sufficient to produce helium line emission. We also discuss numerical results for Be stars with other evolved companions, such as helium stars and neutron stars, and suggest an explanation for the absence of Be-black-hole binaries.