We present a search for nine lepton-number-violating and three lepton-flavor-violating neutral charm decays of the type D^{0}→h^{'-}h^{-}ℓ^{'+}ℓ^{+} and D^{0}→h^{'-}h^{+}ℓ^{'±}ℓ^{∓}, where h and h^{'} represent a K or π meson and ℓ and ℓ^{'} an electron or muon. The analysis is based on 468  fb^{-1} of e^{+}e^{-} annihilation data collected at or close to the ϒ(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)×10^{-7}. The limits are between 1 and 3 orders of magnitude more stringent than previous measurements.