We show how to obtain the ultraviolet central charge from the exact S-
matrix for a wide variety of models with a U(1) symmetry. This is done
by coupling the U(1) current J to a background field. In an N=2 super
conformal theory with J the fermion number current, the OPE of J with
itself and hence the free energy are proportional to c. By deforming t
he supersymmetry into affine su(2)q quantum-group symmetry, this resul
t can be generalized to many U(1) -invariant theories, including the N
=0 and N=1 sine-Gordon models and the SU(2)k WZW models. This provides
a consistency check on a conjectured S-matrix completely independent
of the finite-size effects expressed in terms of dilogarithms resultin
g from the thermodynamic Bethe ansatz.