Universal term in the free energy at a critical point and the conformal anomaly

We show that the leading finite-size correction to lnZ for a two-dimensional system at a conformally invariant critical point on a strip of length L width () is (/6) c (L/), where c is the conformal anomaly. Equivalently, the leading low-temperature correction to the free energy of a one-dimensional quantum system is - (/6) cL (kT) ^2, where v is the effective ``velocity of light. '' The latter formula is used to check recently derived critical theories of spin-s quantum chains against Bethe-Ansatz solutions.