Measurements of the production cross-sections of the Standard Model (SM) Higgs boson (H) decaying into a pair of τ-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 s = 13 TeV, corresponding to an integrated luminosity of 139 fb⁻¹. Leptonic (τ → ℓν_ℓν_τ) and hadronic (τ → hadrons ν_τ) decays of the τ-lepton are considered. All measurements account for the branching ratio of H → ττ and are performed with a requirement |y_H| < 2.5, where y_H is the true Higgs boson rapidity. The cross-section of the pp → H → ττ process is measured to be 2.94 ±0.21(stat)−0.32+0.37(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)−0.67+0.91(syst) pb for gluon-gluon fusion, 0.197 ±0.028(stat)−0.026+0.032(syst) pb for vector-boson fusion, 0.115 ±0.058(stat)−0.040+0.042(syst) pb for vector-boson associated production, and 0.033 ±0.031(stat)−0.017+0.022(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. [Figure not available: see fulltext.]