  Binaries consisting of a pulsar and a black hole (BH) are a holy grail of
astrophysics, both for their significance for stellar evolution and for their
potential application as probes of strong gravity. In spite of extensive
surveys of our Galaxy and its system of globular clusters, no pulsar-black hole
(PSR-BH) binary has been found to date. Clues as to where such systems might
exist are therefore highly desirable. We show that if the central parsec around
Sgr A* harbors a cluster of ~25,000 stellar BHs (as predicted by mass
segregation arguments) and if it is also rich in recycled pulsar binaries (by
analogy with globular clusters), then 3-body exchange interactions should
produce PSR-BHs in the Galactic center. Simple estimates of the formation rate
and survival time of these binaries suggest that a few PSR-BHs should be
present in the central parsec today. The proposed formation mechanism makes
unique predictions for the PSR-BH properties: 1) the binary would reside within
~1 pc of Sgr A*; 2) the pulsar would be recycled, with a period of ~1 to a few
tens of milliseconds, and a low magnetic field B<~10^10 G; 3) the binary would
have high eccentricity, e~0.8, but with a large scatter; and 4) the binary
would be relatively wide, with semi-major axis a_b~0.1 - >~3 AU. The potential
discovery of a PSR-BH binary therefore provides a strong motivation for deep,
high-frequency radio searches for recycled pulsars toward the Galactic center.
