The first measurement of the Higgs boson ($\mathrm{H}$) width is performed based on evidence for off-shell $\mathrm{H}$ production in the final state with two $\mathrm{Z}$ bosons decaying into either four charged leptons ($\mathrm{e}$ or $\mu$), or two charged leptons and two neutrinos. Results are based on data from the CMS experiment at the LHC at a center-of-mass energy of 13~\mathrm{TeV}, corresponding to an integrated luminosity of up to 140~\mathrm{fb}^{-1}. The total size of off-shell $\mathrm{H}$ production beyond the $\mathrm{Z}$ boson pair production threshold is constrained at $95\%$ confidence to be within the interval $[0.0061, 2.0]$ times its standard model (SM) expectation. The scenario with no off-shell production is excluded at a confidence level larger than $99.9\%$ (3.6 standard deviations). The width of the $\mathrm{H}$ boson is then extracted to be \Gamma_{\mathrm{H}} = 3.2_{-1.7}^{+2.4}~\mathrm{MeV}, in agreement with the SM expectation of 4.1~\mathrm{MeV}. The data are also used to set new constraints on anomalous $\mathrm{H}$ boson couplings to massive electroweak vector boson pairs.

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