  A lattice QCD approach to the calculation of the long-distance contributions
to $\epsilon_K$ is presented. This parameter describes indirect CP violation in
$K\to\pi\pi$ decay. While the short-distance contribution to $\epsilon_K$ can
be accurately calculated in terms of standard model parameters and a single
hadronic matrix element, $B_K$, there is a long-distance part which is
estimated to be approximately $5\%$ of the total and is more difficult to
determine. A method for determining this small but phenomenologically important
contribution to $\epsilon_K$ using lattice QCD is proposed and a complete
exploratory calculation of the contribution is presented. This exploratory
calculation uses an unphysical light quark mass corresponding to a 339 MeV pion
mass and an unphysical charm quark mass of 968 MeV, expressed in the
$\overline{\mathrm{MS}}$ scheme at 2 GeV. This calculation demonstrates that
future work should be able to determine this long-distance contribution from
first principles with a controlled error of 10\% or less.
