In this paper, cognitive transmission under quality-of-service (QoS) constraints is studied. In the cognitive radio channel model, it is assumed that the secondary transmitter sends the data at two different fixed rates and power levels, depending on the activity of the primary users, which is determined by channel sensing performed by the secondary users. A state transition model with eight states is constructed to model this cognitive transmission channel. Statistical limitations on the buffer lengths are imposed to take into account the QoS constraints. The maximum throughput under these statistical QoS constraints is identified by finding the effective capacity of the cognitive radio channel. The impact upon the effective capacity of several system parameters, including channel sensing duration, detection threshold, detection and false alarm probabilities, QoS parameters, and fixed transmission rates is investigated. It is noted that the optimal transmission rates are independent of the primary users' activity in the case of perfect channel sensing.