In this work we present a full-duplex (FD) underwater acoustic (UWA) communication system simultaneously transmitting and receiving acoustic signals in the same frequency bandwidth. To simplify the FD hardware, the system exploits a recently designed two-element transducer capable of simultaneously transmitting and receiving signals. The key challenge of implementing an FD system is to cancel the strong self-interference (SI) from the near-end transmitter. By using advanced adaptive filtering algorithms providing high accuracy channel estimates, a high level of SI cancellation can be achieved when the far-end signal is absent. However, the SI channel estimation performance is limited in FD scenarios since the far-end signal acts as an interference. In this paper, we propose an FD UWA communication system which alternates between the SI cancellation and far-end data demodulation. An adaptive Rake combiner with multipath interference cancellation is implemented to improve the demodulation performance in time-varying multipath channels. The performance of the FD UWA system is evaluated in lake experiments. The proposed adaptive Rake combiner with multipath interference cancellation significantly outperforms the conventional Rake combiner in all the experiments. With the new Rake combiner, the detection performance of the proposed FD UWA system is comparable with that of the half-duplex system.