  It is proved that for any $0<\beta<\alpha$, any bounded Ahlfors
$\alpha$-regular space contains a $\beta$-regular compact subset that embeds
biLipschitzly in an ultrametric with distortion at most
$O(\alpha/(\alpha-\beta))$. The bound on the distortion is asymptotically tight
when $\beta\to \alpha$. The main tool used in the proof is a regular form of
the ultrametric skeleton theorem.
