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We study energy-efficient transmission of data with deadline constraints over a time-varying channel. Specifically, the system model consists of a wireless transmitter with controllable transmission rate, time-varying and stochastic channel state, and strict delay constraints on the packets in the queue. While the transmitter can control the rate, the transmission power required depends on the chosen rate and the prevailing channel condition. The objective is to obtain a rate control policy that serves the data within the deadline constraints while minimizing the total energy expenditure. Toward this end, we first introduce the canonical problem of transmitting B units of data by deadline T over a Markov fading channel, and obtain the optimal policy for it using continuous-time stochastic control theory. Using a novel cumulative curves methodology and a decomposition approach, we extend the above setup to consider extensions involving variable deadlines on the packets. Finally, utilizing the analysis we present a heuristic policy for the case of arbitrary packet arrivals to the queue with individual deadline constraints, and give illustrative simulation results for its performance.