  We present the converse to a higher dimensional, scale-invariant version of a
classical theorem of F. and M. Riesz. More precisely, for $n\geq 2$, for an ADR
domain $\Omega\subset \re^{n+1}$ which satisfies the Harnack Chain condition
plus an interior (but not exterior) Corkscrew condition, we show that absolute
continuity of harmonic measure with respect to surface measure on
$\partial\Omega$, with scale invariant higher integrability of the Poisson
kernel, is sufficient to imply uniformly rectifiable of $\partial\Omega$.
