  We present a search for seven lepton-flavor-violating neutral charm decays of
the type $D^{0}\rightarrow X^{0} e^{\pm} \mu^{\mp}$, where $X^{0}$ represents a
$\pi^{0}$, $K^{0}_{\rm S}$, $\bar{K^{*0}}$, $\rho^{0}$, $\phi$, $\omega$, or
$\eta$ meson. The analysis is based on $468$ fb$^{-1}$ of $e^+e^-$ annihilation
data collected at or close to the $\Upsilon(4S)$ resonance with the BaBar
detector at the SLAC National Accelerator Laboratory. No significant signals
are observed, and we establish 90\% confidence level upper limits on the
branching fractions in the range $(5.0 - 22.5)\times 10^{-7}$. The limits are
between one and two orders of magnitude more stringent than previous
measurements.
