This paper reports the measurement of Higgs boson production in association with a $t\bar{t}$ pair in the $H\rightarrow b\bar{b}$ decay channel. The analysis uses 140 fb$^{-1}$ of 13 $\text {TeV}$ proton–proton collision data collected with the ATLAS detector at the Large Hadron Collider. The final states with one or two electrons or muons are employed. An excess of events over the expected background is found with an observed (expected) significance of 4.6 (5.4) standard deviations. The $t\bar{t}H$ cross-section is $\sigma _{t\bar{t}H} = 411~^{+101}_{-92}~\text {fb} = 411 \pm 54(\text {stat.})~^{+85}_{-75}(\text {syst.})~\text {fb}$ for a Higgs boson mass of 125.09 $\text {GeV}$, consistent with the prediction of the Standard Model of $507^{+35}_{-50}$ fb. The cross-section is also measured differentially in bins of the Higgs boson transverse momentum within the simplified template cross-section framework.