  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 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 $\sigma_{t\bar{t}H} =
411^{+101}_{-92}$ fb $= 411 \pm 54( \text{stat.}) ^{+85}_{-75}( \text{syst.})$
fb for a Higgs boson mass of 125.09 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.
