  Fix a constant $C\geq 1$ and let $d=d(n)$ satisfy $d\leq \ln^{C} n$ for every
large integer $n$. Denote by $A_n$ the adjacency matrix of a uniform random
directed $d$-regular graph on $n$ vertices. We show that, as long as
$d\to\infty$ with $n$, the empirical spectral distribution of appropriately
rescaled matrix $A_n$ converges weakly in probability to the circular law. This
result, together with an earlier work of Cook, completely settles the problem
of weak convergence of the empirical distribution in directed $d$-regular
setting with the degree tending to infinity. As a crucial element of our proof,
we develop a technique of bounding intermediate singular values of $A_n$ based
on studying random normals to rowspaces and on constructing a product structure
to deal with the lack of independence between the matrix entries.
