  Sampling edges from a graph in sublinear time is a fundamental problem and a
powerful subroutine for designing sublinear-time algorithms. Suppose we have
access to the vertices of the graph and know a constant-factor approximation to
the number of edges. An algorithm for pointwise $\varepsilon$-approximate edge
sampling with complexity $O(n/\sqrt{\varepsilon m})$ has been given by Eden and
Rosenbaum [SOSA 2018]. This has been later improved by T\v{e}tek and Thorup
[STOC 2022] to $O(n \log(\varepsilon^{-1})/\sqrt{m})$. At the same time,
$\Omega(n/\sqrt{m})$ time is necessary. We close the problem, by giving an
algorithm with complexity $O(n/\sqrt{m})$ for the task of sampling an edge
exactly uniformly.
