Abstract: This Letter reports on a search for off-shell production of the Higgs boson using 139 fb(-1) of pp collision data at root 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 -> 4l and Z Z -> 2l2 nu with l = e or mu. In the ZZ -> 4l 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(-2.5)(+3.3) MeV, and the observed (expected) upper limit on the total width is found to be 10.5 (10.9) MeV at 95% confidence level.