The Hilbert space of a conformal field theory (CFT) is a positive-energy representation of the direct sum of two commuting copies of the Virasoro algebra, and hence decomposes \[ H=\oplus_{h,\bar h\geq 0}V(h,c)\oplus \bar V(\bar h,c) \tag{*} \] where \(V(h,c)\) is the irreducible representation of the Virasoro algebra with highest weight \(h\) and central charge \(c\), the multiplicity \(N_{h\bar h}\in Z_+\) and the bar refers to the second copy of the Virasoro. The partition function \(Z=\sum_{h,\bar h\geq 0}N_{h\bar h}\chi (h,c){\bar \chi}(\bar h,c)\) where \(\chi(h,c)\) is the character of \(V(h,c)\), is holomorphic in the unit disc. The CFT is said to be rational if the matrix \((N_{h\bar h})\) is of finite rank, and modular invariant if \(Z\) is modular invariant. CFT's which are modular invariant and rational include the unitary discrete series and Wess-Zumino-Witten models, among others. In this paper, it is proved that, in any rational, modular invariant CFT, the central charge \(c\) and all the highest weights \(h\), \(\bar h\) which occur are rational numbers.