  The rare Higgs boson decay $H\to Z\gamma$ is forbidden at tree-level. In the
Standard Model, it is loop-mediated through a $W$ boson or a heavy quark. We
analytically compute the QCD correction to the heavy quark loop, confirming
earlier purely numerical results, that were obtained for on-shell
renormalization. The small quark mass expansion of the decay matrix element
contains only single-logarithmic contributions at each perturbative order,
which is in contrast to the double logarithms observed in $H\to \gamma\gamma$.
We investigate the numerical interplay of bottom and top quark contributions,
and the dependence of the result on the renormalization scheme.
