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We consider optimal rate control for energy-efficient data transmission over time-varying (fading) channels with strict deadline constraints. Specifically, the scenario consists of a transmitter with B units of data that must be transmitted by deadline T over a wireless channel. The transmitter can control the transmission rate over time by varying the transmission power subject to expected short-term power limits. The expended power depends on both the (chosen) transmission rate and the present channel condition and the objective is to adapt the rate over time and in response to the changing channel conditions so that the total energy cost is minimized. We present a novel continuous-time formulation of the problem; using stochastic control theory and Lagrangian duality, we obtain explicitly the optimal rate control policy. We then present an illustrative simulation example comparing the energy costs of the optimal and the full power policies.