  We prove that the isometry group $\Iso(\Ur)$ of the universal Urysohn metric
space $\Ur$ equipped with the natural Polish topology is a L\'evy group in the
sense of Gromov and Milman, that is, admits an approximating chain of compact
(in fact, finite) subgroups, exhibiting the phenomenon of concentration of
measure. This strengthens an earlier result by Vershik stating that $\Iso(\Ur)$
has a dense locally finite subgroup.
