We present the result from a precision measurement of the mass of the τ lepton, Mτ, based on 423fb-1 of data recorded at the Υ(4S) resonance with the BABAR detector. Using a pseudomass endpoint method, we determine the mass to be 1776.68±0.12(stat)±0.41(syst)MeV. We also measure the mass difference between the τ+ and τ-, and obtain (Mτ+-Mτ-)/ MAVGτ=(-3.4±1.3(stat)±0.3(syst))×10-4, where MAVGτ is the average value of Mτ+ and Mτ-. © 2009 The American Physical Society.