This Letter describes the observation of the light-by-light scattering process, gamma gamma -> gamma gamma, in Pb + Pb collisions at root S-NN = 5.02 TeV. The analysis is conducted using a data sample corresponding to an integrated luminosity of 1.73 nb(-1), collected in November 2018 by the ATLAS experiment at the LHC. Light-by-light scattering candidates are selected in events with two photons produced exclusively, each with transverse energy E-T(gamma) > 3 GeV and pseudorapidity vertical bar eta(gamma)vertical bar < 2.4, diphoton invariant mass above 6 GeV, and small diphoton transverse momentum and acoplanarity. After applying all selection criteria, 59 candidate events are observed for a background expectation of 12 +/- 3 events. The observed excess of events over the expected background has a significance of 8.2 standard deviations. The measured fiducial cross section is 78 +/- 13(stat) +/- 7(syst) +/- 3(lumi) nb.