  We show that if $n\geq 1$, $\Omega\subset \mathbb R^{n+1}$ is a connected
domain with porous boundary, and $E\subset \partial\Omega$ is a set of finite
and positive Hausdorff $H^{n}$-measure upon which the harmonic measure $\omega$
is absolutely continuous with respect to $H^{n}$, then $\omega|_E$ is
concentrated on an $n$-rectifiable set.
