  We study the classification of immersed constant mean curvature (CMC) spheres
in the homogeneous Riemannian 3-manifold Sol_3, i.e., the only Thurston
3-dimensional geometry where this problem remains open. Our main result states
that, for every H>1/(\sqrt{3}), there exists a unique (up to left translations)
immersed CMC H sphere S_H in Sol_3 (Hopf-type theorem). Moreover, this sphere
S_H is embedded, and is therefore the unique (up to left translations) compact
embedded CMC H surface in Sol_3 (Alexandrov-type theorem). The uniqueness parts
of these results are also obtained for all real numbers H such that there
exists a solution of the isoperimetric problem with mean curvature H.
