  It is known for some time that a random graph $G(n,p)$ contains w.h.p. a
Hamiltonian cycle if $p$ is larger than the critical value $p_{crit}= (\log n +
\log \log n + \omega_n)/n$. The determination of a concrete Hamiltonian cycle
is even for values much larger than $p_{crit}$ a nontrivial task. In this paper
we consider random graphs $G(n,p)$ with $p$ in $\tilde{\Omega}(1/\sqrt{n})$,
where $\tilde{\Omega}$ hides poly-logarithmic factors in $n$. For this range of
$p$ we present a distributed algorithm ${\cal A}_{HC}$ that finds w.h.p. a
Hamiltonian cycle in $O(\log n)$ rounds. The algorithm works in the synchronous
model and uses messages of size $O(\log n)$ and $O(\log n)$ memory per node.
