Abstract: Exclusive production of dielectron pairs, gamma gamma -> e(+) e(-), is studied using L-int = 1.72 nb(-1) of data from ultraperipheral collisions of lead nuclei at root s(NN) = 5.02TeV recorded by the ATLAS detector at the LHC. The process of interest proceeds via photon-photon interactions in the strong electromagnetic fields of relativistic lead nuclei. Dielectron production is measured in the fiducial region defined by following requirements: electron transverse momentum p(T)(e) > 2.5 GeV, absolute electron pseudorapidity |eta(e)| < 2.5, dielectron invariant mass m(ee) > 5 GeV, and dielectron transverse momentum p(T)(ee) < 2 GeV. Differential cross-sections are measured as a function of mee, average peT, absolute dielectron rapidity |y(ee)|, and scattering angle in the dielectron rest frame, | cos theta* |, in the inclusive sample, and also with a requirement of no activity in the forward direction. The total integrated fiducial cross-section is measured to be 215 +/- 1(stat.) (+23)(-20)(syst.) +/- 4(lumi.) μb. Within experimental uncertainties the measured integrated cross-section is in good agreement with the QED predictions from the Monte Carlo programs Starlight and SuperChic, confirming the broad features of the initial photon fluxes. The differential cross-sections show systematic differences from these predictions which are more pronounced at high |y(ee)| and | cos theta* | values.