The rare electromagnetic J/psi -> mu(+)mu(-)mu(+)mu(-) decay is observed with a significance greatly exceeding the discovery threshold, using proton-proton collision data collected by the LHCb experiment during 2016-2018 at a center-of-mass energy of 13 TeV, corresponding to an integrated luminosity of 5.4 fb(-1). The rate of this decay is measured relative to that of the J/psi -> mu(+)mu(-) mode. Using the QED model for the four-muon decay in the efficiency estimation, its branching fraction is determined to be B(J/psi -> mu(+)mu(-)mu(+)mu(-)) = (1.13 +/- 0.10 +/- 0.05 +/- 0.01) x 10(-6), where the uncertainties are statistical, systematic and due to the uncertainty on the branching fraction of the J/psi -> mu(+)mu(-) decay.