  We present a prescription for constructing the monodromy matrix, $\hat{\cal
M}(\omega)$, for $O(d,d)$ invariant string effective actions and derive its
transformation properties under the $T$-duality group. This allows us to
construct $\hat{\cal M}(\omega)$ for new backgrounds, starting from known ones,
which are related by $T$-duality. As an application, we derive the monodromy
matrix for the exactly solvable Nappi-Witten model, both when B=0 and $B\neq
0$.
