  We present a detailed description of the method and results of our
calculation of the kaon B parameter using the Wilson quark action in quenched
QCD at $\beta=5.9-6.5$. The mixing problem of the $\Delta s=2$ four-quark
operators is solved non-perturbatively with full use of chiral Ward identities.
We find $B_K(NDR, 2 GeV)=0.562(64)$ in the continuum limit, which agrees with
the value obtained with the Kogut-Susskind quark action.
