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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.
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