In this paper, we study the propagation of electro-elastic Love waves in a piezoelectric layered solid consisting of a transversely isotropic piezoelectric ceramic film deposited on an isotropic metal substrate. Dispersion relations are obtained in a very concise mathematical form for both electrically open and shorted circuits. Numerical example is given for a PZT-4/steel system. An interesting phenomenon on the fundamental mode of the Love waves in the case of electrically shorted circuit is realized and validated through the discussion on the limit values of phase velocity using the obtained dispersion relations. This validation appears new and is significant to understand the Love wave modes themselves and to the design of acoustic wave devices.