We consider a random access primary network. At the beginning of each time slot, a number of secondary users sense the channel and make an access decision based on the spectrum sensing outcome and the channel state information (CSI). Specifically, the channel is accessed by a secondary transmitter with a probability that depends on both the sensing metric and the gain or signal-to-noise-ratio (SNR) of the channel between the transmitter and its respective receiver. Spectrum sensing operates in a “soft” mode where the sensing metric is used directly rather than making a binary decision concerning primary activity. We consider backlogged secondary users and primary users with infinite queues. The secondary access probabilities are obtained via solving an optimization problem designed to maximize the secondary throughput given a constraint on primary queue stability. The problem is shown to be convex and, hence, the global optimum can be obtained efficiently. Numerical results reveal a significant performance improvement in the secondary throughout with stable primary queues over the use of spectrum sensing with conventional detection or the implementation of sensing alone without making use of the CSI information.