  Binaries containing a stellar-mass black hole and a recycled radio pulsar
have so far eluded detection. We present a focused investigation of the
formation and evolution of these systems in the Galactic disk, highlighting the
factors that limit their numbers and the reasons why they may be extremely
rare. We surmise that the birthrate of black-hole/recycled-pulsar binaries in
the Galactic disk is probably no higher than ~10^{-7}/yr, and may be much less.
Simple arguments regarding common-envelope evolution suggest that these
binaries should have orbital periods less than 10 hr and an average lifetime of
<~10^8 yr before coalescence due to the emission of gravitational radiation. We
expect that fewer than ~10 of these compact, relativistic binaries currently
reside in the Galactic disk, less than 0.1--1% of the number of double neutron
stars. The discovery of two or more black-hole/recycled-pulsar binaries using
current radio telescopes would tightly constrain certain ideas regarding the
evolution of massive stars, dynamical mass transfer, and black-hole formation.
