  We present a spin foam model in which the fundamental ``bubble amplitudes''
(the analog of the one-loop corrections in quantum field theory) are finite as
the cutoff is removed. The model is a natural variant of the field theoretical
formulation of the Barrett-Crane model. As the last, the model is a quantum BF
theory plus an implementation of the constraint that reduces BF theory to
general relativity. We prove that the fundamental bubble amplitudes are finite
by constructing an upper bound, using certain inequalities satisfied by the
Wigner (3n)j-symbols, which we derive in the paper. Finally, we present
arguments in support of the conjecture that the bubble diagrams of the model
are finite at all orders.
