  We prove that given two metrics $g_{+}$ and $g_{-}$ with curvature $\kappa
<-1$ on a closed, oriented surface $S$ of genus $\tau\geq 2$, there exists an
$AdS$ manifold $N$ with smooth, space-like, strictly convex boundary such that
the induced metrics on the two connected components of $\partial N$ are equal
to $g_{+}$ and $g_{-}$. Using the duality between convex space-like surfaces in
$AdS_{3}$, we obtain an equivalent result about the prescription of the third
fundamental form.
