A search for the rare eta -> mu(+)mu(-)mu(+)mu(-) double-Dalitz decay is performed using a sample of proton-proton collisions, collected by the CMS experiment at the CERN LHC with high-rate muon triggers during 2017 and 2018 and corresponding to an integrated luminosity of 101 fb(-1). A signal having a statistical significance well in excess of 5 standard deviations is observed. Using the eta -> mu(+)mu(-) decay as normalization, the branching fraction B(eta -> mu(+)mu(-)mu(+)mu(-)) = [5.0 +/- 0.8(stat) +/- 0.7(syst) +/- 0.7(B-2 mu)] x 10(-9) is measured, where the last term is the uncertainty in the normalization channel branching fraction. This work achieves an improved precision of over 5 orders of magnitude compared to previous results, leading to the first measurement of this branching fraction, which is found to agree with theoretical predictions.