A search for high-mass resonances decaying into a τ-lepton and a neutrino using proton-proton collisions at a center-of-mass energy of s=13  TeV is presented. The full run 2 data sample corresponding to an integrated luminosity of 139  fb-1 recorded by the ATLAS experiment in the years 2015–2018 is analyzed. The τ-lepton is reconstructed in its hadronic decay modes and the total transverse momentum carried out by neutrinos is inferred from the reconstructed missing transverse momentum. The search for new physics is performed on the transverse mass between the τ-lepton and the missing transverse momentum. No excess of events above the Standard Model expectation is observed and upper exclusion limits are set on the W′→τν production cross section. Heavy W′ vector bosons with masses up to 5.0 TeV are excluded at 95% confidence level, assuming that they have the same couplings as the Standard Model W boson. For nonuniversal couplings, W′ bosons are excluded for masses less than 3.5–5.0 TeV, depending on the model parameters. In addition, model-independent limits on the visible cross section times branching ratio are determined as a function of the lower threshold on the transverse mass of the τ-lepton and missing transverse momentum.