  We analyse the eigenvectors of the adjacency matrix of the Erd\H{o}s-R\'enyi
graph $\mathbb G(N,d/N)$ for $\sqrt{\log N} \ll d \lesssim \log N$. We show the
existence of a localized phase, where each eigenvector is exponentially
localized around a single vertex of the graph. This complements the completely
delocalized phase previously established in [arXiv:2005.14180]. For large
enough $d$, we establish a mobility edge by showing that the localized phase
extends up to the boundary of the delocalized phase. We derive explicit
asymptotics for the localization length up to the mobility edge and
characterize its divergence near the phase boundary.
  The proof is based on a rigorous verification of Mott's criterion for
localization, comparing the tunnelling amplitude between localization centres
with the eigenvalue spacing. The first main ingredient is a new family of
global approximate eigenvectors, for which sharp enough estimates on the
tunnelling amplitude can be established. The second main ingredient is a lower
bound on the spacing of approximate eigenvalues. It follows from an
anticoncentration result for these discrete random variables, obtained by a
recursive application of a self-improving anticoncentration estimate due to
Kesten.
