  Measurements of the Standard Model Higgs boson decaying into a $b\bar{b}$
pair and produced in association with a $W$ or $Z$ boson decaying into leptons,
using proton-proton collision data collected between 2015 and 2018 by the ATLAS
detector, are presented. The measurements use collisions produced by the Large
Hadron Collider at a centre-of-mass energy of $\sqrt{s} = $13 TeV,
corresponding to an integrated luminosity of 139 fb$^{-1}$. The production of a
Higgs boson in association with a $W$ or $Z$ boson is established with observed
(expected) significances of 4.0 (4.1) and 5.3 (5.1) standard deviations,
respectively. Cross-sections of associated production of a Higgs boson decaying
into bottom quark pairs with an electroweak gauge boson, $W$ or $Z$, decaying
into leptons are measured as a function of the gauge boson transverse momentum
in kinematic fiducial volumes. The cross-section measurements are all
consistent with the Standard Model expectations, and the total uncertainties
vary from 30% in the high gauge boson transverse momentum regions to 85% in the
low regions. Limits are subsequently set on the parameters of an effective
Lagrangian sensitive to modifications of the $WH$ and $ZH$ processes as well as
the Higgs boson decay into $b\bar{b}$.
