  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 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) x 10^{-8} at 90% confidence level.
