The existence of the dielectric constant epsilon is investigated for fluid mixtures of rigid polar molecules. The investigation is performed using the functional-derivative formalism for mixtures, and is closely analogous to that previously carried out for pure dipolar fluids (J. Chem. Phys. 68, 5199 (1978)). Sufficient conditions for the existence of epsilon are obtained in terms of the direct correlation function matrix c/sub alphabeta/(12). It is found that epsilon exists if c/sub alphabeta/(12) depends only on relative positions and orientations, and becomes asymptotic to -theta/sub alphabeta/(12)/kT at long range, where theta/sub alphabeta/(12) is the dipole--dipole potential between a molecule of species ..cap alpha.. and one of species ..beta... An expression for epsilon in terms of the short-range total correlation function matrix emerges automatically from the development. This expression is equivalent to an earlier result obtained by a different method. Expressions for epsilon in terms of c/sub alphabeta/(12) are derived for axially symmetric molecules and for molecules of arbitrary symmetry. In the former case, the expression involves the inverse of an N/sub c/ x N/sub c/ matrix, where N/sub c/ is the number of components in the mixture. This expression facilitates the evaluation of epsilon in the mean spherical approximation. For molecules of arbitrary symmetry, the expression for epsilon in terms of c/sub alphabeta/(12) involves the inverse of an N/sub c/ x N/sub c/ supermatrix, each element of which is a 3 x 3 matrix.