  We report the observation of the rare charm decay $D^0\rightarrow
K^-\pi^+e^+e^-$, based on $468$ fb$^{-1}$ of $e^+e^-$ annihilation data
collected at or close to the center-of-mass energy of the $Y(4S)$ resonance
with the BaBar detector at the SLAC National Accelerator Laboratory. We find
the branching fraction in the invariant mass range $0.675< m(e^+e^-) < 0.875
GeV/c^2$ of the electron-positron pair to be ${\cal B}(D^0\rightarrow
K^-\pi^+e^+e^-) = (4.0\pm0.5\pm0.2\pm0.1) \times 10^{-6}$, where the first
uncertainty is statistical, the second systematic, and the third due to the
uncertainty in the branching fraction of the decay $D^0\rightarrow
K^-\pi^+\pi^+\pi^-$ used as a normalization mode. The significance of the
observation corresponds to 9.7 standard deviations including systematic
uncertainties. This result is consistent with the recently reported
$D^0\rightarrow K^-\pi^+\mu^+\mu^-$ branching fraction, measured in the same
invariant mass range, and with the value expected in the Standard Model. In a
set of regions of $m(e^+e^-)$ where long-distance effects are potentially
small, we determine a 90% confidence level upper limit on the branching
fraction ${\cal B}(D^0\rightarrow K^-\pi^+e^+e^-) < 3.1 \times 10^{-6}$.
