A search for the neutrinoless, lepton-flavor violating decay of the tau lepton into three charged leptons has been performed using 376 fb{-1} of data collected at an e{+}e{-} center-of-mass energy around 10.58 GeV with the BABAR detector at the SLAC PEP-II storage rings. In all six decay modes considered, the numbers of events found in data are compatible with the background expectations. Upper limits on the branching fractions are set in the range (4-8)x10{-8} at 90% confidence level.