This paper presents an energy-efficient power allocation for a multicarrier link over a frequency-selective fading channel with a delay-outage probability constraint. The power adaptation maximizes the system energy efficiency (EE), formulated as the ratio of the achieved effective capacity (EC) to the total expenditure power, including both transmission power and rate-independent circuit power. We prove that this objective function is quasi-concave in the transmission power, and derive the global optimum solution using fractional programming. Based on the obtained solution, we develop a power adaptation algorithm consisting of two steps: (i) establishing the optimum average power level corresponding to the maximum achievable EE with no transmit power constraint, and then (ii) for a given power constraint, jointly distributing the power over time and frequency based on the constraint and the optimum power level found in the first step. Analytical results show that the proposed EE-based power allocation has a structure similar to the allocation that maximizes the EC, but with a different cutoff threshold. Our proposed joint EE-optimal power allocation provides significant EE gains over both the joint spectral-efficient and independent-subcarrier EE-based power allocation schemes, where the rate-energy tradeoff becomes more pronounced with higher frequency selectivity.