We consider the influence of an ω-dependent ionic dielectric constant ϵ(ω) on the properties of a superconductor. Assuming that the pairing interaction is proportional to ϵ2 we have solved the Eliashberg equations for this case, both for imaginary and real frequencies. The interaction potential depends on a coupling constant λ and on a longitudinal phonon frequency Ω. The dielectric constant is assumed to be independent of wavevector q, and to depend on frequency through the expression: ϵ(ω) = (ω2 - ω2long)/(ω2 - ω2trans), where ωlong, ωtrans are the frequencies of optical phonons of the dielectric. We find that along the imaginary frequency axis (but not for real frequencies) the weighted phonon propagator can be modeled by an appropriate choice of a cutoff frequency and an effective coupling constant. The influence of ϵ(ω) on Tc, the gap δ(ω), and the renormalization function Z(ω) are studied and it is found that these quantities increase significantly with the dielectric constant.