Wireless systems that carry delay-sensitive information (such as speech and/or video signals) typically transmit with fixed data rates, but may occasionally suffer from transmission outages caused by the random nature of the fading channels. If the transmitter has instantaneous channel state information (CSI) available, it can compensate for a significant portion of these outages by utilizing power allocation. In a conventional dual-hop bidirectional amplify-and-forward (AF) relaying system, the relay already has instantaneous CSI of both links available, as this is required for relay gain adjustment. We therefore develop an optimal power allocation strategy for the relay, which adjusts its instantaneous output power to the minimum level required to avoid outages, but only if the required output power is below some cutoff level; otherwise, the relay is silent in order to conserve power and prolong its lifetime. The proposed scheme is proven to minimize the system outage probability, subject to an average power constraint at the relay and fixed output powers at the end nodes.