An energy scan near the τ pair production threshold has been performed using the BESIII detector. About 24 pb-1 of data, distributed over four scan points, were collected. This analysis is based on τ pair decays to ee, eμ, eh, μμ, μh, hh, eρ, μρ and πρ final states, where h denotes a charged π or K. The mass of the τ lepton is measured from a maximum likelihood fit to the τ pair production cross-section data to be mτ = (1776.91 ± 0.12-0.13+0.10) MeV=c2, which is currently the most precise value in a single measurement.