  Let M be a closed orientable Seifert fibered 3-manifold with a hyperbolic
base 2-orbifold, or equivalently, admitting a geometry modeled on H^2 \times R
or the universal cover of SL(2,R). Our main result is that the connected
component of the identity map in the diffeomorphism group Diff(M) is either
contractible or homotopy equivalent to the circle, according as the center of
the fundamental group of M is trivial or infinite cyclic. Apart from the
remaining case of non-Haken infranilmanifolds, this completes the homeomorphism
classifications of Diff(M) and of the space of Seifert fiberings of M for all
compact orientable aspherical 3-manifolds. We also prove that when the base
orbifold of M is hyperbolic with underlying manifold the 2-sphere with three
cone points, the inclusion from the isometry group Isom(M) to Diff(M) is a
homotopy equivalence.
