A structure theorem is
proven for closed Euclidean 3-dimensional cone manifolds with all cone angles
greater than 2π and cone locus a link (no vertices) which allows one to deduce
precisely when such a manifold is homotopically atoroidal, and to construct its
characteristic submanifold (torus decomposition) when it is not. A by-product of this
structure theorem is the result that any Seifert-fibered submanifold of such a
manifold admits a fibration with fibers parallel to the cone locus. This structure
theorem is applied to several examples arising as branched covers over universal
links.