For a parallel-plate or coaxial transmission line, a dielectric layer of sufficiently high permittivity placed in the space between the conductors, but adjacent to neither, can result in an attenuation coefficient which is significantly lower than that of the air-filled line. The dominant mode in these lines is shown to be a TM mode with no lower cutoff frequency. The expressions for the attenuation coefficient, derived by a perturbational technique, are applicable for any range of line dimensions and dielectric-layer permittivity and thickness.