  We develop and demonstrate techniques needed to compute the long distance
contribution to the $K_{L}$-$K_{S}$ mass difference, $\Delta M_K$, in lattice
QCD and carry out a first, exploratory calculation of this fundamental
quantity. The calculation is performed on 2+1 flavor, domain wall fermion,
$16^3\times32$ configurations with a 421 MeV pion mass and an inverse lattice
spacing $1/a=1.73$ GeV. We include only current-current operators and drop all
disconnected and double penguin diagrams. The short distance part of the mass
difference in a 2+1 flavor calculation contains a quadratic divergence cut off
by the lattice spacing. Here, this quadratic divergence is eliminated through
the GIM mechanism by introducing a valence charm quark. The inclusion of the
charm quark makes the complete calculation accessible to lattice methods
provided the discretization errors associated with the charm quark can be
controlled. The long distance effects are discussed for each parity channel
separately. While we can see a clear signal in the parity odd channel, the
signal to noise ratio in the parity even channel is exponentially decreasing as
the separation between the two weak operators increases. We obtain a mass
difference $\Delta M_K$ which ranges from $6.58(30)\times 10^{-12}$ MeV to
$11.89(81)\times 10^{-12}$ MeV for kaon masses varying from 563 MeV to 839 MeV.
Extensions of these methods are proposed which promise accurate results for
both $\Delta M_K$ and $\epsilon_K$, including long distance effects.
