We prove the orbifold theorem in the cyclic case: If O is a compact oriented irreducible atoroidal 3-orbifold whose ramification locus is a non-empty submanifold, then O is geometric, i.e. it has a hyperbolic, a Euclidean or a Seifert fibred structure. This theorem implies Thurston's geometrization conjecture for compact orientable irreducible three-manifolds having a non-free symmetry.