  We report the results of a calculation of the K --> pi pi matrix elements
relevant for the $\DIhalf$ rule and $\epe$ in quenched lattice QCD using domain
wall fermions at a fixed lattice spacing $a^{-1} \sim 2$ GeV. Working in the
three-quark effective theory, where only the u, d and s quarks enter and which
is known perturbatively to next-to-leading order, we calculate the lattice K
--> pi and K --> |0> matrix elements of dimension six, four-fermion operators.
Through lowest order chiral perturbation theory these yield K --> pi pi matrix
elements, which we then normalize to continuum values through a
non-perturbative renormalization technique. For the ratio of isospin amplitudes
|A_0|/|A_2| we find a value of $25.3 \pm 1.8$ (statistical error only) compared
to the experimental value of 22.2, with individual isospin amplitudes 10-20%
below the experimental values. For $\epe$, using known central values for
standard model parameters, we calculate $(-4.0 \pm 2.3) \times 10^{-4}$
(statistical error only) compared to the current experimental average of $(17.2
\pm 1.8) \times 10^{-4}$. Because we find a large cancellation between the I =
0 and I = 2 contributions to $\epe$, the result may be very sensitive to the
approximations employed. Among these are the use of: quenched QCD, lowest order
chiral perturbation theory and continuum perturbation theory below 1.3 GeV. We
have also calculated the kaon B parameter, B_K and find $B_{K,\bar{MS}}(2 {\rm
GeV}) = 0.532(11)$. Although currently unable to give a reliable systematic
error, we have control over statistical errors and more simulations will yield
information about the effects of the approximations on this first-principles
determination of these important quantities.
