Rare or forbidden decays of the $D^0$ meson are an excellent probe for physics beyond the Standard Model. In this talk, we report the observation of the rare charm decay $D^0 \rightarrow K^- \pi^+ e^+ e^-$ with a 9.7$\sigma$ significance. In addition, we present a search for nine lepton-number-violating and three lepton-flavor-violating neutral charm decays of the type $D^0 \rightarrow h^{'-} h^- \ell^{'+}\ell^+$ and $D^0 \rightarrow h^{'-} h^+\ell^{\pm}\ell^{\mp}$, where h and h$'$ represent a K or $\pi$ meson. Furthermore, a search for seven lepton-number-violating decays of the type $D^0 \rightarrow X^0 e^{\pm} \mu^{\mp}$ , with $X^0$ a $\pi^0$, $K_S^0$, $K^{*0}$, $\rho^0$, $\phi$, $\omega$, or $\eta$ meson, is also presented. The results are based on 468 fb$^{-1}$ of $e^{+}e^{-}$ collision data collected at or close to the $\Upsilon(4S)$ resonance with the \babar{} detector at the SLAC National Accelerator Laboratory.