  This work is devoted to the study of deformations of hyperbolic cone
structures under the assumption that the lengths of the singularity remain
uniformly bounded over the deformation. Given a sequence (M_{i},p_{i}) of
pointed hyperbolic cone-manifolds with topological type (M,{\Sigma}), where M
is a closed, orientable and irreducible 3-manifold and {\Sigma} an embedded
link in M. Assuming that the lengths of the singularity remain uniformly
bounded, we prove that either the sequence M_{i} collapses and M is Seifert
fibered or a Sol manifold, or the sequence M_{i} does not collapse and in this
case a subsequence of (M_{i},p_{i}) converges to a complete Alexandrov space of
dimension 3 endowed with a hyperbolic metric of finite volume on the complement
of a finite union of quasi-geodesics. We apply this result to a conjecture of
Thurston and to the case where {\Sigma} is a small link in M.
