  We present non-perturbative renormalization factors for $\Delta S=2$
four-quark operators in quenched domain-wall QCD using the Schroedinger
functional method. Non-perturbative renormalization factor for $B_K$ is
evaluated at hadronic scale. Combined with the non-perturbative RG running
obtained by the Alpha collaboration, our result yields renormalization factor
which converts lattice bare $B_K$ to the renormalization group invariant one.
We apply the renormalization factor to bare $B_K$ previously obtained by the
CP-PACS collaboration with the quenched domain-wall QCD(DWQCD). We compare our
result with previous ones obtained by perturbative renormalization factors,
different renormalization schemes or different quark actions. We also show that
chiral symmetry breaking effects in the renormalization factor are numerically
small.
