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Given a weakly o-minimal structure 
${\cal M}$ and its o-minimal completion 
$\bar{{\cal M}}$, we first associate to 
$\bar{{\cal M}}$ a canonical language and then prove that Th
$\left( {\cal M} \right)$ determines 
$Th\left( {\bar{{\cal M}}} \right)$. We then investigate the theory of the pair 
$\left( {\bar{{\cal M}},{\cal M}} \right)$ 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 definable open subset of 
${\bar{M}^n}$ is already definable in 
$\bar{{\cal M}}$.
We give an example of a weakly o-minimal structure interpreting 
$\bar{{\cal M}}$ and show that it is not elementarily equivalent to any reduct of an o-minimal trace.
