  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 cross-section of the $pp\to H\to\tau\tau$ 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 $3.17\pm0.09~ \mbox{pb}$. 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.
