We present a measurement of the τ-lepton mass using a sample of about 175 million e+e−→τ+τ− events collected with the Belle II detector at the SuperKEKB e+e− collider at a center-of-mass energy of 10.579 GeV. This sample corresponds to an integrated luminosity of 190 fb−1. We use the kinematic edge of the τ pseudomass distribution in the decay τ−→π−π+π−ντ and measure the τ mass to be 1777.09±0.08±0.11 MeV/c2, where the first uncertainty is statistical and the second systematic. This result is the most precise to date.