Measurements of the production of a Higgs boson in association with a $t\bar{t}$ pair in the $H\rightarrow b\bar{b}$ decay channel are presented. The analysis uses 140~fb$^{-1}$ of 13 TeV proton--proton collision data collected with the ATLAS detector at the Large Hadron Collider. Final states with one or two electrons or muons are considered for signal selection. 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 $\sigma_{t\bar{t}H}= 411^{+101}_{-92}=411\pm 54(\text{stat.})^{+85}_{-75}(\text{syst.}) \text{fb}$ for a Higgs boson mass of 125.09 GeV, consistent with the prediction of the Standard Model of $507^{+35}_{-50}$ \text{fb}. The cross-section is also measured differentially in bins of the Higgs boson transverse momentum within the simplified template cross-section framework.