  The rare decays of a kaon into a pion and a charged lepton/antilepton pair
proceed via a flavour changing neutral current and therefore may only be
induced beyond tree level in the Standard Model. This natural suppression makes
these decays sensitive to the effects of potential New Physics. The CP
conserving $K\to\pi \ell^+\ell^-$ decay channels however are dominated by a
single photon exchange; this involves a sizeable long-distance hadronic
contribution which represents the current major source of theoretical
uncertainty. Here we outline our methodology for the computation of the
long-distance contributions to these rare decay amplitudes using lattice QCD
and present the numerical results of the first exploratory studies of these
decays in which all but the disconnected diagrams are evaluated. The domain
wall fermion ensembles of the RBC and UKQCD collaborations are used, with a
pion mass of $M_{\pi}\sim 430\,\mathrm{MeV}$ and a kaon mass of $M_{K}\sim
625\,\mathrm{MeV}$. In particular we determine the form factor, $V(z)$, of the
$K^+\to\pi^+\ell^+\ell^-$ decay from the lattice at small values of
$z=q^2/M_{K}^{2}$, obtaining $V(z)=1.37(36),\, 0.68(39),\, 0.96(64)$ for the
three values of $z=-0.5594(12),\, -1.0530(34),\, -1.4653(82)$ respectively.
