CERN-LHC, ATLAS. A search for pair-produced scalar or vector leptoquarks (LQs) decaying into a $b$-quark and a $\tau$-lepton is presented using the full LHC Run 2 (2015-2018) data sample of 139 fb$^{\text{-1}}$ collected with the ATLAS detector in proton-proton collisions at a centre-of-mass energy of $\sqrt{s}$ = 13 TeV. Events in which at least one $\tau$-lepton decays hadronically are considered, and multivariate discriminants are used to extract the signals. No significant deviations from the Standard Model expectation are observed and 95% confidence-level upper limits on the production cross-section are derived as a function of m$_\text{LQ}$ and branching ratio B into a $\tau$-lepton and $b$-quark. For scalar LQs, masses below 1460 GeV are excluded assuming B = 100%, while for vector LQs the corresponding limit is 1650 GeV (1910 GeV) in the minimal-coupling (Yang-Mills, YM) scenario. Basic event selection: - ==0(1) electron or muon for $\tau_\text{lep}\tau_\text{had}$ ($\tau_\text{had}\tau_\text{had}$) region: $p_T$ thresholds are corresponding trigger, in fiducial volume, isolated from jets - ==1(2) hadronically decaying $\tau$s ($\tau_{had}$s) for $\tau_\text{lep}\tau_\text{had}$ ($\tau_\text{had}\tau_\text{had}$) region: $p_T$ > 100 GeV (thresholds are corresponding trigger) for $\tau_\text{lep}\tau_\text{had}$ ($\tau_\text{had}\tau_\text{had}$) region, in fiducial volume, isolated from jets - >=2 small-R jets, with 1 or 2 $b$-tagged - Opposite charge electron/muon/$\tau_{had}$s and $\tau_{had}$s - MMC mass from 2 leptons ($m_{MMC}$) should be not 40 < $m_{MMC}$ < 150 GeV (Z veto) - Missing transverse momentum > 100 GeV - Scalar sum of $p_T$ ($s_T$) > 600 GeV See paper for more detail.