  We propose a new, discretized model for the study of 3+1-dimensional
canonical quantum gravity, based on the classical $SL(2,\C)$-connection
formulation. The discretization takes place on a topological $N^3$- lattice
with periodic boundary conditions. All operators and wave functions are
constructed from one-dimensional link variables, which are regarded as the
fundamental building blocks of the theory. The kinematical Hilbert space is
spanned by polynomials of certain Wilson loops on the lattice and is manifestly
gauge- and diffeomorphism- invariant. The discretized quantum Hamiltonian $\hat
H$ maps this space into itself. We find a large sector of solutions to the
discretized Wheeler-DeWitt equation $\hat H\psi=0$, which are labelled by
single and multiple Polyakov loops. These states have a finite norm with
respect to a natural scalar product on the space of holomorphic
$SL(2,\C)$-Wilson loops. We also investigate the existence of further solutions
for the case of the $1^3$-lattice. - Our results provide for the first time a
rigorous, regularized framework for studying non-perturbative quantum gravity.
