  We calculate the amplitudes involved in the heavy baryon nonleptonic decay
$\Lambda_b \rar \Lambda J/\psi$ using perturbative QCD factorization theorem,
which are expressed as convolutions of hard $b$ quark decay amplitudes with the
$\Lambda_b$ baryon, $\Lambda$ baryon and $J/\psi$ meson distribution
amplitudes. It is found that nonfactorizable contributions dominate over
factorizable ones. Because of soft cancellation in pairs of nonfactorizable
diagrams, the $\Lambda_b \rar \Lambda J/\psi$ decay is characterized by a large
scale, such that perturbative QCD is applicable. Employing the distribution
amplitudes determined in our previous works and from QCD sum rules, we derive
the branching ratio $B(\Lambda_b \rar \Lambda J/\psi)=(1.7 \sim 5.3)\times
10^{-4}$ in agreement with data. We predict an asymmetry parameter $\alpha=
-0.17 \sim -0.14$ associated with the anisotropic angular distribution of the
$\Lambda$ baryons produced in polarized $\Lambda_b$ baryon decays.
