  We report on a search for the flavor-changing neutral current decays $\Dz\to
e^+e^-$ and $\Dz\to\mu^+\mu^-$, and the lepton-flavor violating decay $\Dz\to
e^\pm\mu^\mp$. The measurement is based on $122 {fb}^{-1}$ of data collected by
the \babar detector at the PEP-II asymmetric $e^+e^-$ collider. No evidence is
found for any of the decays. The upper limits on the branching fractions, at
the 90 % confidence level, are $1.2\times 10^{-6}$ for $\Dz\to e^+e^-$,
$1.3\times 10^{-6}$ for $\Dz\to\mu^+\mu^-$, and $8.1\times 10^{-7}$ for $\Dz\to
e^\pm\mu^\mp$.
