  This Letter presents the measurement of the fiducial and differential
cross-sections of the electroweak production of a $Z \gamma$ pair in
association with two jets. The analysis uses 140 fb$^{-1}$ of LHC proton-proton
collision data taken at $\sqrt{s}$=13 TeV recorded by the ATLAS detector during
the years 2015-2018. Events with a $Z$ boson candidate decaying into either an
$e^+e^-$ or $\mu^+ \mu^-$ pair, a photon and two jets are selected. The
electroweak component is extracted by requiring a large dijet invariant mass
and a large rapidity gap between the two jets and is measured with an observed
and expected significance well above five standard deviations. The fiducial $pp
\rightarrow Z \gamma jj$ cross-section for the electroweak production is
measured to be 3.6 $\pm$ 0.5 fb. The total fiducial cross-section that also
includes contributions where the jets arise from strong interactions is
measured to be $16.8^{+2.0}_{-1.8}$ fb. The results are consistent with the
Standard Model predictions. Differential cross-sections are also measured using
the same events and are compared with parton-shower Monte Carlo simulations.
Good agreement is observed between data and predictions.
