We study the interaction between a fleet of electric self-driving vehicles servicing on-demand transportation requests (referred to as Autonomous Mobility-on-Demand, or AMoD, system) and the electric power network. We propose a joint model that captures the coupling between the two systems stemming from the vehicles' charging requirements, capturing time-varying customer demand, battery depreciation, and power transmission constraints. First, we show that the model is amenable to efficient optimization. Then, we prove that the socially optimal solution to the joint problem is a general equilibrium if locational marginal pricing is used for electricity. Finally, we show that the equilibrium can be computed by selfish transportation and generator operators (aided by a non-profit ISO) without sharing private information. We assess the performance of the approach and its robustness to stochastic fluctuations in demand through case studies and agent-based simulations. Collectively, these results provide a first-of-a-kind characterization of the interaction between AMoD systems and the power network, and shed additional light on the economic and societal value of AMoD.