  We study 2-dimensional submanifolds of the space
${\mathbb{L}}({\mathbb{H}}^3)$ of oriented geodesics of hyperbolic 3-space,
endowed with the canonical neutral K\"ahler structure. Such a surface is
Lagrangian iff there exists a surface in ${\mathbb{H}}^3$ orthogonal to the
geodesics of $\Sigma$.
  We prove that the induced metric on a Lagrangian surface in
${\mathbb{L}}({\mathbb{H}}^3)$ has zero Gauss curvature iff the orthogonal
surfaces in ${\mathbb{H}}^3$ are Weingarten: the eigenvalues of the second
fundamental form are functionally related. We then classify the totally null
surfaces in ${\mathbb{L}}({\mathbb{H}}^3)$ and recover the well-known
holomorphic constructions of flat and CMC 1 surfaces in ${\mathbb{H}}^3$.
