  We calculate the complete NLO QCD corrections to loop-induced $gg \rightarrow
ZZ$ production including full top-quark mass effects. The two-loop virtual
corrections are obtained by combining analytic results for the massless,
Higgs-mediated, and one-loop factorizable contributions with numerically
computed amplitudes containing the top-quark mass. We show that the choice of
subtraction scheme for the virtual contribution impacts the precision with
which the virtual contribution must be evaluated in order to obtain
sufficiently precise phenomenological predictions. For direct production
through a massive top-quark loop, we observe that the relative NLO corrections
are large. The direct massive and Higgs-mediated contributions individually
increase relative to the massless production at high diboson invariant mass,
but interfere destructively with each other. At the Large Hadron Collider, the
NLO corrections to the gluon channel give a sizable contribution to the $pp \to
ZZ+X$ cross-section at N${}^3$LO.
