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Starting from Ooguri's construction for $BF$ theory in three (and four)dimensions, we show how to construct a well defined theory with an infinitenumber of degrees of freedom. The spin network states that are kept invariantby the evolution operators of the theory are exact solutions of the Hamiltonianconstraint of quantum gravity proposed by Thiemann. The resulting theory is thefirst example of a well defined, finite, consistent, spin-foam based theory ina situation with an infinite number of degrees of freedom. Since it solves thequantum constraints of general relativity it is also a candidate for a theoryof quantum gravity. It is likely, however, that the solutions constructedcorrespond to a spurious sector of solutions of the constraints. The richnessof the resulting theory makes it an interesting example to be analyzed byforthcoming techniques that construct the semi-classical limit of spin networkquantum gravity.Comment: 4 pages, Revtex, 6 figures included with psfig, to appear in PR

Finite spin-foam-based theory of three- and four-dimensional quantum gravity