Measurements of the production cross-sections of the Standard Model (SM) Higgs boson (H) decaying into a pair of tau -leptons are presented. The measurements use data collected with the ATLAS detector from pp collisions produced at the Large Hadron Collider at a centre-of-mass energy of p root s = 13TeV, corresponding to an integrated luminosity of 139 fb-1. Leptonic ( tau -> l upsilon(l)upsilon(tau)) and hadronic ( tau -> hadrons upsilon tau) decays of the tau -lepton are considered. All measurements account for the branching ratio of H -> tau tau and are performed with a requirement |yH| < 2.5, where yH is the true Higgs boson rapidity. The cross-section of the pp -> H -> tau tau process is measured to be 2.94 +/- 0.21(stat)+ 0.37 - 0.32(syst) pb, in agreement with the SM prediction of 3.17 +/- 0.09 pb. Inclusive cross-sections are determined separately for the four dominant production modes: 2.65 +/- 0.41(stat)+ 0.91 - 0.67(syst) pb for gluon-gluon fusion, 0.197 +/- 0.028(stat)+ 0.032 - 0.026(syst) pb for vectorboson fusion, 0.115 +/- 0.058(stat)+ 0.042 - 0.040(syst) pb for vector-boson associated production, and 0.033 +/- 0.031(stat)+ 0.022 - 0.017(syst) pb for top-quark pair associated production. Measurements in exclusive regions of the phase space, using the simplified template cross-section framework, are also performed. All results are in agreement with the SM predictions.