  We consider the adjacency matrix $A$ of the Erd\H{o}s-R\'enyi graph on $N$
vertices with edge probability $d/N$. For $(\log \log N)^4 \ll d \lesssim \log
N$, we prove that the eigenvalues near the spectral edge form asymptotically a
Poisson process and the associated eigenvectors are exponentially localized. As
a corollary, at the critical scale $d \asymp \log N$, the limiting distribution
of the largest nontrivial eigenvalue does not match with any previously known
distribution. Together with [arXiv:2005.14180], our result establishes the
coexistence of a fully delocalized phase and a fully localized phase in the
spectrum of $A$. The proof relies on a three-scale rigidity argument, which
characterizes the fluctuations of the eigenvalues in terms of the fluctuations
of sizes of spheres of radius 1 and 2 around vertices of large degree.
