  Given a weakly o-minimal structure $\mathcal M$ and its o-minimal completion
$\bar {\mathcal M}$, we first associate to $\bar {\mathcal M}$ a canonical
language and then prove that $Th(\mathcal M)$ determines $Th(\bar {\mathcal
M})$. We then investigate the theory of the pair $\mathcal M^P=(\bar {\mathcal
M};M)$ in the spirit of the theory of dense pairs of o-minimal structures, and
prove, among other results, that it is near model complete, and every $\mathcal
M^P$-definable open subset of $\bar M^n$ is already definable in $\bar
{\mathcal M}$. We give an example of a weakly o-minimal structure which
interprets $\bar {\mathcal M}$ and show that it is not elementarily equivalent
to any reduct of an o-minimal trace.
