  The KTeV E799 experiment has conducted a search for the rare decay
$K_{L}\to\pi^{0}\pi^{0}\gamma$ via the topology
$K_{L}\to\pi^{0}\pi^{0}_D\gamma$ (where $\pi^0_D\to\gamma e^+e^-$). Due to Bose
statistics of the $\pi^0$ pair and the real nature of the photon, the
$K_{L}\to\pi^{0}\pi^{0}\gamma$ decay is restricted to proceed at lowest order
by the CP conserving direct emission (DE) of an E2 electric quadrupole photon.
The rate of this decay is interesting theoretically since chiral perturbation
theory predicts that this process vanishes at level $O(p^4)$. Therefore, this
mode probes chiral perturbation theory at $O(p^6)$. In this paper we report a
determination of an upper limit of $2.43\times 10^{-7}$ (90% CL) for
$K_{L}\to\pi^{0}\pi^{0}\gamma$. This is approximately a factor of 20 lower than
previous results.
