  We report on a calculation of $B_K$ with domain wall fermion action in
quenched QCD. Simulations are made with a renormalization group improved gauge
action at $\beta=2.6$ and 2.9 corresponding to $a^{-1}\approx 2$GeV and 3GeV.
Effects due to finite fifth dimensional size $N_5$ and finite spatial size
$N_\sigma$ are examined in detail. Matching to the continuum operator is made
perturbatively at one loop order. We obtain $B_K(\mu = 2 {GeV})= 0.5746(61)$,
where the error is statistical only, as an estimate of the continuum value in
the $\msbar$ scheme with naive dimensional regularization. This value is
smaller but consistent with $B_K(\mu = 2 {GeV})= 0.628(42)$ obtained by the
JLQCD Collaboration using the Kogut-Susskind quark action. Results for strange
quark mass are also reported.
