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In this paper, a novel cognitive multiple-access strategy in the presence of a cooperating relay is proposed. Exploiting an important phenomenon in wireless networks, source burstiness, the cognitive relay utilizes the periods of silence of the terminals to enable cooperation. Therefore, no extra channel resources are allocated for cooperation and the system encounters no bandwidth losses. Two protocols are developed to implement the proposed multiple-access strategy. The maximum stable throughput region and the delay performance of the proposed protocols are characterized. The results reveal that the proposed protocols provide significant performance gains over conventional relaying strategies such as selection and incremental relaying, specially at high spectral efficiency regimes. The rationale is that the lossless bandwidth property of the proposed protocols results in a graceful degradation in the maximum stable throughput with increasing the required rate of communication. On the other hand, conventional relaying strategies suffer from catastrophic performance degradation because of their inherent bandwidth inefficiency that results from allocating specific channel resources for cooperation at the relay. The analysis reveals that the throughput region of the proposed strategy is a subset of its maximum stable throughput region, which is different from random access, where both regions are conjectured to be identical.