Inclusive and differential fiducial cross sections of the Higgs boson are measured in the H rightarrow ZZ^{*} rightarrow 4ell (ell = e,mu ) decay channel. The results are based on proton-proton collision data produced at the Large Hadron Collider at a centre-of-mass energy of 13 TeV and recorded by the ATLAS detector from 2015 to 2018, equivalent to an integrated luminosity of 139 hbox {fb}^{-1}. The inclusive fiducial cross section for the H rightarrow ZZ^{*} rightarrow 4ell process is measured to be sigma _mathrm {fid} = 3.28 ,{pm }, 0.32 fb, in agreement with the Standard Model prediction of sigma _mathrm {fid, SM} = 3.41 pm 0.18 fb. Differential fiducial cross sections are measured for a variety of observables which are sensitive to the production and decay of the Higgs boson. All measurements are in agreement with the Standard Model predictions. The results are used to constrain anomalous Higgs boson interactions with Standard Model particles.