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 ($\tau\rightarrow\ell\nu_{\ell}\nu_{\tau}$) and hadronic (\tau\rightarrow\text{hadrons}\, \nu_{\tau}) decays of the $\tau$-lepton are considered. All measurements account for the branching ratio of $H\rightarrow\tau\tau$ and are performed with a requirement $|y_H|<2.5$, where $y_H$ is the true Higgs boson rapidity. The cross-section of the $pp\rightarrow H\rightarrow\tau\tau$ process is measured to be 2.94 \pm 0.21 \text{(stat)} ^{+\,0.37}_{-\,0.32} \text{(syst)} pb, in agreement with the SM prediction of $3.17\pm0.09$ pb. Inclusive cross-sections are determined separately for the four dominant production modes: 2.65 \pm 0.41 \text{(stat)} ^{+\,0.91}_{-\,0.67} \text{(syst)} pb for gluon-gluon fusion, 0.197 \pm 0.028 \text{(stat)} ^{+\,0.032}_{-\,0.026} \text{(syst)} pb for vector-boson fusion, 0.115 \pm 0.058 \text{(stat)} ^{+\,0.042}_{-\,0.040} \text{(syst)} pb for vector-boson associated production, and 0.033 \pm 0.031 \text{(stat)} ^{+\,0.022}_{-\,0.017} \text{(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.

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