  Let $E\subset \mathbb{R}^{n+1}$, $n\ge 2$, be an Ahlfors-David regular set of
dimension $n$. We show that the weak-$A_\infty$ property of harmonic measure,
for the open set $\Omega:= \mathbb{R}^{n+1}\setminus E$, implies uniform
rectifiability of $E$.
