A search for the decay of a Standard Model Higgs boson into a $b\bar{b}$ pair when produced in association with a $W$ or $Z$ boson has been performed with the ATLAS detector. Data were collected in proton-proton collisions from Run 2 of the Large Hadron Collider at a centre-of-mass energy of 13 TeV, corresponding to an integrated luminosity of 13.2 fb$^{-1}$. Final states are considered that contain 0, 1 and 2 charged leptons (electrons or muons), targeting the decays: $Z \rightarrow \nu \nu$, $W \rightarrow \ell \nu$, and $Z \rightarrow \ell \ell$. For $m_H = 125$~GeV the ratio of the measured signal strength to the SM expectation is found to be $\mu = 0.21 ^{+0.36}_{-0.35}\mathrm{(stat.)} \pm 0.36 \mathrm{(syst.)}$. This corresponds to an observed significance of 0.42 standard deviations compared with an expected sensitivity of 1.94. The analysis procedure has been validated by measuring the yield of $(W/Z)Z$ with $Z \rightarrow b\bar{b}$, where the ratio of the observed yield to that expected in the Standard Model was found to be $0.91 \pm 0.17 \mathrm{(stat.)} ^{+ 0.32}_{-0.27} \mathrm{(syst.)}$, corresponding to a significance of 3.0 standard deviations compared to an expected significance of 3.2.

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