This Letter reports on a search for off-shell production of the Higgs boson using 139 fb−1 of pp collision data at 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 ZZ→2ℓ2ν with ℓ=e or μ. 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.