This paper considers the use of full-duplex technology in cognitive radio networks to allow secondary users to sense the presence of primary users and transmit data simultaneously. This is the main advantage over half-duplex radios. In such networks, the so-called sensing-throughput trade-off exists due to the fact that while a higher transmit power results in higher secondary network throughput, sensing performance is degraded by the self-interference at the full-duplex transceiver. This paper presents a novel deep reinforcement learning-based joint spectrum sensing and power control algorithm for downlink communications in a cognitive small cell. The proposed algorithm can adapt to the unknown radio environment to transmit data opportunistically to the secondary users while avoiding interference to the primary network. Simulation results show that our algorithm achieves better performance than the traditional energy detection-based sensing method and performs close to a genie-aided method with the optimal spectrum utilization, especially in the high-SNR regime.