Link adaptation is an important issue in the design of cognitive radio networks, which aims at making efficient use of system resources. In this paper, we propose and investigate a joint adaptive modulation and power allocation algorithm in cognitive radio networks. Specifically, the modulation scheme and transmit power are adjusted adaptively according to channel conditions, interference limit and target signal-to-interference-plus-noise ratio (SINR). As such the total power consumption of cognitive users (CUs) is minimized while keeping both the target SINR of CUs and interference to primary user (PU) at an acceptable level. Simulation results are provided to show that the proposed algorithm achieves a significant gain in power saving.