A novel probabilistic radio resource allocation algorithm is proposed for code-division multiple-access cognitive radio networks. Our method incorporates a sensing-based hybrid access- and interference-limited spectrum sharing strategy to maximize the achievable spectral efficiency of cognitive users (CUs) while guaranteeing the quality-of-service (QoS) requirements of licensed services. Specifically, the enhanced performance for CUs is achieved through optimal joint power and rate adaptation to the time-varying multiuser interference and frequency-selective fading channel conditions. We show that this scheme can improve the average spectral efficiency of CUs, while satisfying the peak and average received-interference constraints at the primary receiver.