  Let $\log^Cn\le d\le n/2$ for a sufficiently large constant $C>0$ and let
$A_n$ denote the adjacency matrix of a uniform random $d$-regular directed
graph on $n$ vertices. We prove that as $n$ tends to infinity, the empirical
spectral distribution of $A_n$, suitably rescaled, is governed by the Circular
Law. A key step is to obtain quantitative lower tail bounds for the smallest
singular value of additive perturbations of $A_n$.
