We report on a search for the flavor-changing neutral current decays D0-->e(+)e(-) and D0-->mu(+)mu(-), and the lepton-flavor violating decay D0-->e(+/-)mu(-/+). The measurement is based on 122 fb(-1) of data collected by the BABAR detector at the SLAC 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 1.2x10(-6) for D0-->e(+)e(-), 1.3x10(-6) for D0-->mu(+)mu(-), and 8.1x10(-7) for D0-->e(+/-)mu(-/+).