  Let $E\subset \mathbb{R}^{n+1}$, $n\ge 1$, be a uniformly rectifiable set of
dimension $n$. We show $E$ that has big pieces of boundaries of a class of
domains which satisfy a 2-sided corkscrew condition, and whose connected
components are all chord-arc domains (with uniform control of the various
constants). As a consequence, we deduce that $E$ has big pieces of sets for
which harmonic measure belongs to weak-$A_\infty$
