  We study conditions under which the Hausdorff quasi-uniformity ${\mathcal
U}_H$ of a quasi-uniform space $(X,{\mathcal U})$ on the set ${\mathcal
P}_0(X)$ of the nonempty subsets of $X$ is bicomplete.
  Indeed we present an explicit method to construct the bicompletion of the
$T_0$-quotient of the Hausdorff quasi-uniformity of a quasi-uniform space. It
is used to find a characterization of those quasi-uniform $T_0$-spaces
$(X,{\mathcal U})$ for which the Hausdorff quasi-uniformity
$\widetilde{{\mathcal U}}_H$ of their bicompletion
$(\widetilde{X},{\widetilde{\mathcal U}})$ on ${\mathcal P}_0(\widetilde{X})$
is bicomplete.
