The Standard Model decay of the Higgs boson to leptons has been observed in the decay to a pair of $\tau$-leptons. The cross section for $H\rightarrow\tau\tau$ is measured to $\sigma^{\mathrm{obs}}_{H\rightarrow\tau\tau} = 2.90 \pm 0.21\,(\mathrm{stat.})\, {}^{+0.37}_{-0.32}\,(\mathrm{syst.})\, \mathrm{pb}$ using Higgs boson decays into two $\tau$-leptons in multiple Higgs production channels. The result is in agreement with the Standard Model prediction of $3.15\pm 0.09\,\mathrm{pb}$. The sensitivity of the analysis is improved using machine-learning techniques. The analysis uses the full Run~2 proton-proton collision dataset corresponding to an integrated luminosity of $139\,\mathrm{fb}^{-1}$ recorded at a centre-of-mass energy $\sqrt{s} = 13\,\mathrm{TeV}$ with the ATLAS detector at the LHC.