Increasing the energy autonomy of a hopping one-legged robot is studied in this paper. For a particular passive gait, of all those possible, the energy dissipated per unit length of travel is shown to be less than for any other gait. This optimal gait is identified analytically, by exploiting the commonly used SLIP model to simplify real robot dynamics. Both mechanical and electrical losses are considered. The accuracy of the optimal gait analytical prediction is evaluated by a numerical analysis of a realistic robot model. Finally, restrictions imposed on executing the optimal gait due to motor limitations are studied.