  Pulsar timing observations have revealed companions to neutron stars that
include other neutron stars, white dwarfs, main-sequence stars, and planets. We
demonstrate that the correlated and apparently stochastic residual times of
arrival from the millisecond pulsar B1937+21 are consistent with the signature
of an asteroid belt having a total mass less than approximately 0.05 Earth
masses. Unlike the solar system's asteroid belt, the best fit pulsar asteroid
belt extends over a wide range of radii, consistent with the absence of any
shepherding companions. We suggest that any pulsar that has undergone
accretion-driven spin-up and subsequently evaporated its companion may harbor
orbiting asteroid mass objects. The resulting timing variations may
fundamentally limit the timing precision of some of the other millisecond
pulsars. Observational tests of the asteroid belt model include identifying
periodicities from individual asteroids, which are difficult; testing for
statistical stationarity that become possible when observations are conducted
over a longer observing span; and searching for reflected radio emission.
