  Starting from Ooguri's construction for $BF$ theory in three (and four)
dimensions, we show how to construct a well defined theory with an infinite
number of degrees of freedom. The spin network states that are kept invariant
by the evolution operators of the theory are exact solutions of the Hamiltonian
constraint of quantum gravity proposed by Thiemann. The resulting theory is the
first example of a well defined, finite, consistent, spin-foam based theory in
a situation with an infinite number of degrees of freedom. Since it solves the
quantum constraints of general relativity it is also a candidate for a theory
of quantum gravity. It is likely, however, that the solutions constructed
correspond to a spurious sector of solutions of the constraints. The richness
of the resulting theory makes it an interesting example to be analyzed by
forthcoming techniques that construct the semi-classical limit of spin network
quantum gravity.
