  Exclusive production of dielectron pairs, $\gamma\gamma\rightarrow e^+e^-$,
is studied using $\mathcal{L}_\mathrm{int}=1.72\; \mathrm{nb^{-1}}$ of data
from ultraperipheral collisions of lead nuclei at $\sqrt{s_{_{\text{NN}}}} =
5.02$ TeV recorded by the ATLAS detector at the LHC. The process of interest
proceeds via photon-photon interaction in the strong electromagnetic fields of
relativistic lead nuclei. Dielectron production is measured in the fiducial
region defined by following requirements: electron transverse momentum,
$p_{\textrm{T}}^{e} > 2.5$ GeV, absolute electron pseudorapidity, $|\eta^{e}| <
2.5$, invariant mass of the dielectron system, $m_{ee} > 5$ GeV, and transverse
momentum of the dielecton pair, $p_{\textrm{T}}^{ee} < 2$ GeV. Differential
cross-sections are measured as a function of $m_{ee}$, average
$p_{\textrm{T}}^{e}$, absolute rapidity of the dielectron system, $|y_{ee}|$,
and scattering angle in the dielectron rest frame, $|\cos\theta^*|$ in the
inclusive sample, and also under the requirement of no activity in the forward
direction. The total integrated fiducial cross-section is measured to be $215
\pm 1 \text{(stat.)} ^{+23}_{-20} \text{(syst.)} \pm 4 \text{(lumi.)}\; \mu$b.
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 with these predictions
which are more pronounced at high $|y_{ee}|$ and $|\cos\theta^*|$ values.
