  We present a measurement of $B(\pi^0 \rightarrow e^+e^- \gamma)/B(\pi^0
\rightarrow \gamma\gamma)$, the Dalitz branching ratio, using data taken in
1999 by the E832 KTeV experiment at Fermi National Accelerator Laboratory. We
use neutral pions from fully reconstructed $K_L$ decays in flight; the
measurement is based on about 60 thousand $K_L \rightarrow \pi^0\pi^0\pi^0
\rightarrow \gamma\gamma~\gamma\gamma~e^+e^-\gamma$ decays. We normalize to
$K_L \rightarrow \pi^0\pi^0\pi^0 \rightarrow 6\gamma$ decays. We find $B(\pi^0
\rightarrow e^+e^- \gamma)/B(\pi^0 \rightarrow \gamma\gamma)$ $(m_{e^+e^-}$ >
15 MeV/$c^2)$ = $[3.920 \pm 0.016(stat) \pm 0.036 (syst)] \times 10^{-3}$.
Using the Mikaelian and Smith prediction for the $e^+e^-$ mass spectrum, we
correct the result to the full $e^+e^-$ mass range. The corrected result is
$B(\pi^0 \rightarrow e^+e^- \gamma)/B(\pi^0 \rightarrow \gamma\gamma) = [1.1559
\pm 0.0047(stat) \pm 0.0106 (syst)]$%. This result is consistent with previous
measurements and the uncertainty is a factor of three smaller than any previous
measurement.
