  We develop chiral perturbation theory for baryons in quenched QCD. Quenching
(the elimination of diagrams containing virtual quark loops) is achieved by
extending the Lagrangian method of Bernard and Golterman, and is implemented in
a theory where baryons are treated as fixed velocity sources. Our method
requires that the octet baryons be represented by a three index tensor rather
than by the usual matrix field. We calculate the leading non-analytic
corrections to the masses of octet and decuplet baryons. In QCD these are
proportional to $M_\pi^3$. We find that quenching alters the $M_\pi^3$ terms,
but does not completely remove them. In addition, we find non-analytic
contributions to baryon masses proportional to $M_\pi$ and $M_\pi^2 \log
M_\pi$. These terms, which are artifacts of quenching, dominate over the
$M_\pi^3$ terms for sufficiently small quark masses. We also point out various
pecularities of the quenched theory, most notably that the $\Delta$ baryon can
decay (if kinematically allowed), in the sense that its two point function will
be dominated at long Euclidean times by a nucleon plus pion intermediate state.
