  We present a search for nine lepton-number-violating and three
lepton-flavor-violating neutral charm decays of the type $D^0\rightarrow
h^{\prime -} h^{-}\ell^{\prime +} \ell^{+}$ and $D^0\rightarrow h^{\prime -}
h^{+}\ell^{\prime\pm} \ell^{\mp}$, where $h$ and $h^{\prime}$ represent a $K$
or $\pi$ meson and $\ell$ and $\ell^{\prime}$ an electron or muon. The analysis
is based on $468$ fb$^{-1}$ of $e^+e^-$ annihilation data collected at or close
to the $Y(4S)$ resonance with the BaBar detector at the SLAC National
Accelerator Laboratory. No significant signal is observed for any of the twelve
modes and we establish 90% confidence level upper limits on the branching
fractions in the range $(1.0 - 30.6)\times 10^{-7}$. The limits are between one
and three orders of magnitude times more stringent than previous measurements.
