Many scenarios of physics beyond the standard model predict the existence of new gauge singlets, which might be substantially lighter than the weak scale. The experimental constraints on additional scalars with masses in the MeV to GeV range could be significantly weakened if they interact predominantly with leptons rather than quarks. At an e+e- collider, such a leptophilic scalar (φL) would be produced predominantly through radiation from a τ lepton. We report herein a search for e+e-→τ+τ-φL, φL→ℓ+ℓ- (ℓ=e, μ) using data collected by the BABAR experiment at SLAC. No significant signal is observed, and we set limits on the φL coupling to leptons in the range 0.04<mφL<7.0 GeV. These bounds significantly improve upon the current constraints, excluding almost entirely the parameter space favored by the observed discrepancy in the muon anomalous magnetic moment below 4 GeV at 90% confidence level.