This paper presents an energy-efficient power allocation strategy for Nakagami-m flat-fading channels with a delay-outage probability constraint. The operating input transmit power value is limited to P-max. The energy efficiency (EE), expressed in units of b/J/Hz, is represented as the ratio of the effective capacity to the sum of transmission power (P-t) and circuit power (P-c). Since the EE-maximization objective function is quasi-concave, a unique global maximum exists. By using fractional programming, we develop an EE-optimal power allocation strategy that consists of two steps: 1) obtaining the power level (P) over bar (un), at which the maximum EE can be achieved, and 2) distributing the power optimally based on the minimum of P-max and (P) over bar (un). We prove that while (P) over bar (un) monotonically increases with P-c, the maximum achievable EE is a monotonically decreasing function of P-c. The analysis further allows us to derive the EE of three important cases: non-fading channels, extremely stringent delay-limited systems, and systems with no delay constraints. Simulation results confirm analytical derivations and further show the effects of the circuit power, fading duration, and fading severeness on the achievable EE and effective capacity of a delay-limited fading channel.