  This Letter reports on a search for off-shell production of the Higgs boson
using 139 $\textrm{fb}^{-1}$ of $pp$ collision data at $\sqrt{s}=$ 13 TeV
collected by the ATLAS detector at the Large Hadron Collider. The signature is
a pair of $Z$ bosons, with contributions from both the production and
subsequent decay of a virtual Higgs boson and the interference of that process
with other processes. The two observable final states are $ZZ\rightarrow 4\ell$
and $ZZ\rightarrow 2\ell2\nu$ with $\ell = e$ or $\mu$. In the $ZZ\rightarrow
4\ell$ final state, a dense Neural Network is used to enhance analysis
sensitivity with respect to matrix element-based discrimination. The
background-only hypothesis is rejected with an observed (expected) significance
of 3.3 (2.2) standard deviations, representing experimental evidence for
off-shell Higgs boson production. Assuming that no new particles enter the
production of the virtual Higgs boson, its total width can be deduced from the
measurement of its off-shell production cross-section. The measured total width
of the Higgs boson is $4.5^{+3.3}_{-2.5}$ MeV, and the observed (expected)
upper limit on the total width is found to be 10.5 (10.9) MeV at 95% confidence
level.
