  We present the calculation of next-to-next-to-leading order (NNLO)
corrections in perturbative QCD for the production of a Higgs boson decaying
into a pair of bottom quarks in association with a leptonically decaying weak
vector boson: $\mathrm{pp} \to V \mathrm{H} + X \to
\ell\bar{\ell}\;\mathrm{b\bar{b}} + X$. We consider the corrections to both the
production and decay sub-processes, retaining a fully differential description
of the final state including off-shell propagators of the Higgs and vector
boson. The calculation is carried out using the antenna subtraction formalism
and is implemented in the NNLOJET framework. Clustering and identification of
$\mathrm{b}$-jets is performed with the flavour-$k_t$ algorithm and results for
fiducial cross sections and distributions are presented for the LHC at
$\sqrt{s}=13\;\text{TeV}$. We assess the residual theory uncertainty by varying
the production and decay scales independently and provide scale uncertainty
bands in our results, yielding percent-level accurate predictions for
observables in this Higgs production mode computed at NNLO. Confronting a
na\"ive perturbative expansion of the cross section against the customary
re-scaling procedure to a fixed branching ratio reveals that starting from
NNLO, the latter could be inadequate in estimating missing higher-order effects
through scale variations.
