  We prove global rigidity for compact hyperbolic and spherical
cone-3-manifolds with cone-angles $\leq \pi$ (which are not Seifert fibered in
the spherical case), furthermore for a class of hyperbolic cone-3-manifolds of
finite volume with cone-angles $\leq \pi$, possibly with boundary consisting of
totally geodesic hyperbolic turnovers. To that end we first generalize our
local rigidity result to the setting of hyperbolic cone-3-manifolds of finite
volume as above. We then use the geometric techniques developed by Boileau,
Leeb and Porti to deform the cone-manifold structure to a complete non-singular
or a geometric orbifold structure, where global rigidity holds due to
Mostow-Prasad rigidity in the hyperbolic case, resp. a result of de Rham in the
spherical case. This strategy has already been implemented successfully by
Kojima in the compact hyperbolic case if the singular locus is a link using
Hodgson-Kerckhoff local rigidity.
