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 $ \sqrt{s} $ = 13 TeV, corresponding to an integrated luminosity of 139 fb$^{−1}$. Leptonic (τ → ℓν$_{ℓ}$ν$_{τ}$) 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{\left(\mathrm{stat}\right)}_{-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{\left(\mathrm{stat}\right)}_{-0.67}^{+0.91} $(syst) pb for gluon-gluon fusion, 0.197 ±$ 0.028{\left(\mathrm{stat}\right)}_{-0.026}^{+0.032} $(syst) pb for vector-boson fusion, 0.115 ±$ 0.058{\left(\mathrm{stat}\right)}_{-0.040}^{+0.042} $(syst) pb for vector-boson associated production, and 0.033 ±$ 0.031{\left(\mathrm{stat}\right)}_{-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.