  Let D(n,p) be the random directed graph on n vertices where each of the
n(n-1) possible arcs is present independently with probability p. A celebrated
result of Frieze shows that if $p\ge(\log n+\omega(1))/n$ then D(n,p) typically
has a directed Hamilton cycle, and this is best possible. In this paper, we
obtain a strengthening of this result, showing that under the same condition,
the number of directed Hamilton cycles in D(n,p) is typically
$n!(p(1+o(1)))^{n}$. We also prove a hitting-time version of this statement,
showing that in the random directed graph process, as soon as every vertex has
in-/out-degrees at least 1, there are typically $n!(\log n/n(1+o(1)))^{n}$
directed Hamilton cycles.
