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In a recent paper, we demonstrated the minimization of instantaneous energy in a two-way relay network with digital network coding at the relay, through the optimization of the time-sharing fractions for each mode of transmission. In that work, the channel gains were assumed to be changing so slowly with time that queue stability required instantaneous channel throughput to be no smaller than the average packet arrival rate. In this paper, we obtain a water-filling solution that minimizes the long-term average (or ergodic) energy required to deliver any given pair of required rates while maintaining queue stability at all three nodes, in a fading channel. We also provide a detailed analysis of the queues at the relay using a random scheduling method that closely approximates the theoretical design of the deterministic algorithm, through a two-dimensional Markov chain model.