  A search for the Higgs boson decaying into a photon and a pair of electrons
or muons with an invariant mass $m_{\ell\ell} < 30$ GeV is presented. The
analysis is performed using 139 fb$^{-1}$ of proton-proton collision data,
produced by the LHC at a centre-of-mass energy of 13 TeV and collected by the
ATLAS experiment. Evidence for the $H \rightarrow \ell \ell \gamma$ process is
found with a significance of 3.2$\sigma$ over the background-only hypothesis,
compared to an expected significance of 2.1$\sigma$. The best-fit value of the
signal strength parameter, defined as the ratio of the observed signal yield to
the one expected in the Standard Model, is $\mu = 1.5 \pm 0.5$. The Higgs boson
production cross-section times the $H \rightarrow\ell\ell\gamma$ branching
ratio for $m_{\ell\ell} <$ 30 GeV is determined to be 8.7 $^{+2.8}_{-2.7}$ fb.
