Abstract 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.