A search for the lepton-flavor-violating decay of the tau into three charged leptons has been performed using 91.5 fb(-1) of data collected at an e(+)e(-)center-of-mass energy around 10.58 GeV with the BABAR detector at the SLAC storage ring PEP-II. 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 (1-3)x10(-7) at 90% confidence level.