A measurement of the mass of the W boson is presented based on proton–proton collision data recorded in 2011 at a centre-of-mass energy of 7 TeV with the ATLAS detector at the LHC, and corresponding to 4.6fb-1 of integrated luminosity. The selected data sample consists of 7.8×106 candidates in the W→μν channel and 5.9×106 candidates in the W→eν channel. The W -boson mass is obtained from template fits to the reconstructed distributions of the charged lepton transverse momentum and of the W boson transverse mass in the electron and muon decay channels, yielding mW=80370±7(stat.)±11(exp. syst.)±14(mod. syst.)MeV=80370±19MeV, where the first uncertainty is statistical, the second corresponds to the experimental systematic uncertainty, and the third to the physics-modelling systematic uncertainty. A measurement of the mass difference between the W+ and W- bosons yields mW+-mW-=-29±28 MeV.