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A search for pair-produced scalar or vector leptoquarks decaying into a $b$-quark and a $\tau$-lepton is presented using the full LHC Run 2 (2015-2018) data sample of 139 fb$^{-1}$ 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 leptoquark mass and branching ratio $B$ into a $\tau$-lepton and $b$-quark. For scalar leptoquarks, masses below 1460 GeV are excluded assuming $B=100$%, while for vector leptoquarks the corresponding limit is 1650 GeV (1910 GeV) in the minimal-coupling (Yang-Mills) scenario.