  A lattice QCD calculation of the kaon $B$ parameter $B_K$ is carried out with
the Wilson quark action in the quenched approximation at $\beta=6/g^2=5.9-6.5$.
The mixing problem of the $\Delta s=2$ four-quark operators is solved
non-perturbatively with full use of chiral Ward identities employing four
external quarks with an equal off-shell momentum in the Landau gauge. This
method, without invoking any effective theory, enables us to construct the weak
four-quark operators exhibiting good chiral behavior. Our results for $B_K$
with the non-perturbative mixing coefficients show small scaling violation
beyond the lattice cut-off $a^{-1}\sim 2.5 $GeV. Our estimate concludes
$B_K(NDR, 2 GeV)=0.69(7)$ at $a^{-1}=2.7-4.3$GeV, which agrees with the value
obtained with the Kogut-Susskind quark action. For comparison we also calculate
$B_K$ with one-loop perturbative mixing coefficients. While this yields
incorrect values at finite lattice spacing, a linear extrapolation to the
continuum limit as a function of $a$ leads to a result consistent with those
obtained with the Ward identity method.
