A new representation 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 the new formalism of Ashtekar (1986). A closed-form general solution is given for the diffeomorphism constraints, and a large class of solutions is given for the full set of constraints. These are classified by the knot and link classes of the spatial three-manifold.