We consider optimal rate-control for energy-efficient transmission of data, over a time-varying channel, with packet-deadline constraints. Specifically, the problem scenario consists of a wireless transmitter with units of data that must be transmitted by deadline over a fading channel. The transmitter can control the transmission rate over time and the required instantaneous power depends on the chosen rate and the present channel condition, with limits on short-term average power consumption. The objective is to obtain the optimal rate-control policy that minimizes the total energy expenditure while ensuring that the deadline constraint is met. Using a continuous-time stochastic control formulation and a Lagrangian duality approach, we explicitly obtain the optimal policy and show that it possesses a very simple and intuitive form. Finally, we present an illustrative simulation example comparing the energy costs of the optimal policy with the full power policy.