  A report is presented on our continued effort to elucidate the continuum
limit of $B_K$ using the quenched Kogut-Susskind quark action. By adding to our
previous simulations one more point at $\beta = 6.65$ employing a $56^3\times
96$ lattice, we now confirm the expected $O(a^2)$ behavior of $B_K$ with the
Kogut-Susskind action. A simple continuum extrapolation quadratic in $a$ leads
to $B_K$(NDR, 2 GeV) = 0.598(5). As our final value of $B_K$ in the continuum
we present $B_K$(NDR, 2 GeV)=0.628(42), as obtained by a fit including an
$\alpha_{\bar{MS}}(1/a)^2$ term arising from the lattice-continuum matching
with the one-loop renormalization.
