Using approximately 350 million {tau}{sup +}{tau}{sup -} pair events recorded with the BaBar detector at the Stanford Linear Accelerator Center between 1999 and 2006, a search has been made for neutrinoless, lepton-flavor violating tau decays to three lighter leptons. All six decay modes consistent with conservation of electric charge and energy have been considered. With signal selection efficiencies of 5-12%, we obtain 90% confidence level upper limits on the branching fraction {Beta}({tau} {yields} {ell}{ell}{ell}) in the range (4-8) x 10{sup -8}.