  We analyse the rare kaon decays $K_S \to \gamma\gamma$ and $K_S \to
\gamma\ell^+\ell^-$ $(\ell = e \mbox{ or } \mu)$ in a dispersive framework in
which the weak Hamiltonian carries momentum. Our analysis extends predictions
from lowest order $SU(3)_L\times SU(3)_R$ chiral perturbation theory
($\chi$PT$_3$) to fully account for effects from final-state interactions, and
is free from ambiguities associated with extrapolating the kaon off-shell.
Given input from $K_S \to \pi\pi$ and $\gamma\gamma^{(*)}\to\pi\pi$, we solve
the once-subtracted dispersion relations numerically to predict the rates for
$K_S \to \gamma\gamma$ and $K_S \to \gamma\ell^+\ell^-$. In the leptonic modes,
we find sizeable corrections to the $\chi$PT$_3$ predictions for the integrated
rates.
