A new represenatation for quantum general relativity is described, which is defined in terms of functionals of sets of loops in three-space. In this representation exact solutions of the quantum constraints may be obtained. This result is related to the simplification of the constraints in Ashtekar's new formalism. We give in closed form the general solution of the diffeomorphisms constraint and a large class of solutions of the full set of constraints. These are classified by the knot and link classes of the spatial three-manifold.