In this paper, an adaptive cooperation diversity scheme with best-relay selection is proposed for multiple-relay cognitive radio networks to improve the performance of secondary transmissions while ensuring the quality of service (QoS) of primary transmissions. Exact closed-form expressions of the outage probability of secondary transmissions, referred to as secondary outage probability, are derived under the constraint of satisfying a required outage probability of primary transmissions (primary outage probability) for both the traditional non-cooperation and the proposed adaptive cooperation schemes over Rayleigh fading channels. Numerical and simulation results show that, with a guaranteed primary outage probability, a floor of the secondary outage probability occurs in high signal-to-noise ratio (SNR) regions. Moreover, the outage probability floor of the adaptive cooperation scheme is lower than that of the non-cooperation scenario, which illustrates the advantage of the proposed ...