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 $B_K(mu = 2 GeV)= 0.5746(61)(191)$, where the first error is statistical and the second error represents an estimate of scaling violation and ${cal O}(alpha^2)$ errors in the renormalization factor added in quadrature, as an estimate of the continuum value in the $msbar$ scheme with naive dimensional regularization. This value is consistent, albeit somewhat small, with $B_K(mu = 2 {GeV})= 0.628(42)$ obtained by the JLQCD Collaboration using the Kogut-Susskind quark action. Results for light quark masses are also reported.
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