  Given a closed orientable Euclidean cone 3-manifold C with cone angles less
than or equal to pi, and which is not almost product, we describe the space of
constant curvature cone structures on C with cone angles less than pi. We
establish a regeneration result for such Euclidean cone manifolds into
spherical or hyperbolic ones and we also deduce global rigidity for Euclidean
cone structures.
