  We carry out the quantization of the full type I and II Bianchi models
following the non-perturbative canonical quantization program. These
homogeneous minisuperspaces are completely soluble, i.e., it is possible to
obtain the general solution to their classical equations of motion in an
explicit form. We determine the sectors of solutions that correspond to
different spacetime geometries, and prove that the parameters employed to
describe the different physical solutions define a good set of coordinates in
the phase space of these models. Performing a transformation from the Ashtekar
variables to this set of phase space coordinates, we endow the reduced phase
space of each of these systems with a symplectic structure. The symplectic
forms obtained for the type I and II Bianchi models are then identified as
those of the cotangent bundles over ${\cal L}^+_{(+,+)}\times S^2\times S^1$
and ${\cal L}^+_{(+,+)}\times S^1$, respectively. We construct a closed
*-algebra of Dirac observables in each of these reduced phase spaces, and
complete the quantization program by finding unitary irreducible
representations of these algebras. The real Dirac observables are represented
in this way by self-adjoint operators, and the spaces of quantum physical
states are provided with a Hilbert structure.
