  Consider a 3$-$dimensional manifold $N$ obtained by gluing a finite number of
ideal hyperbolic tetrahedra via isometries along their faces. By varying the
isometry type of each tetrahedron but keeping fixed the gluing pattern we
define a space $\mathcal{T}$ of complete hyperbolic metrics on $N$ with cone
singularities along the edges of the tetrahedra. We prove that $\mathcal{T}$ is
homeomorphic to a Euclidean space and we compute its dimension. By means of
examples, we examine if the elements of $% \mathcal{T}$ are uniquely determined
by the angles around the edges of $N.$
