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 pages, 6 figures, accepted by ApJ, minor changes