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 \sqrt{s}=13\,\text{TeV}, corresponding to an integrated luminosity of 139\,\text{fb}^{-1}. Leptonic (\tau\to\ell\nu_{\ell}\,\nu_{\tau}) and hadronic ($\tau\to\text{hadrons }\nu_{\tau}$) decays of the $\tau$ lepton are considered. All measurements account for the branching ratio of $H\to\tau\tau$ and are performed with a requirement $|y_H|<2.5$, where $y_H$ is the true Higgs boson rapidity. The production cross-section of the pp\,\to\,H\,\to\,\tau\tau process is measured to be 2.90 $\pm$ 0.21 (stat) ^{+\,0.37}_{-\,0.32}(syst) pb. Inclusive cross-sections are determined separately for the four dominant production modes: 2.7 $\pm$ 0.4 (stat) ^{+\,0.9}_{-\,0.6}(syst) pb for the gluon-gluon fusion, 0.196 ^{+\,0.028}_{-\,0.027}(stat) ^{+\,0.032}_{-\,0.025}(syst) pb for the vector-boson fusion, 0.11 $\pm$ 0.06 (stat) $\pm$ 0.04 (syst) pb for the vector boson associated production, and 0.048 ^{+\,0.033}_{-\,0.029}(stat) ^{+\,0.036}_{-\,0.029}(syst) pb for the 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.

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