  This paper presents a search for exotic decays of the Higgs boson into a pair
of new pseudoscalar particles, $H\rightarrow aa$, where one pseudoscalar decays
into a $b$-quark pair and the other decays into a $\tau$-lepton pair, in the
mass range $12\leq m_{a}\leq 60$ GeV. The analysis uses $pp$ collision data at
$\sqrt{s} = 13$ TeV collected with the ATLAS detector at the LHC, corresponding
to an integrated luminosity of 140 ${fb}^{-1}$. No significant excess above the
Standard Model (SM) prediction is observed. Assuming the SM Higgs boson
production cross-section, the search sets upper limits at 95% confidence level
on the branching ratio of Higgs bosons decaying into $b\bar{b}\tau^+\tau^-$,
$\mathcal{B}(H \rightarrow aa \rightarrow b\bar{b}\tau^+\tau^-)$, between 2.2%
and 3.9% depending on the pseudoscalar mass.
