This paper proposes a cooperative nonorthogonal multiple access (CNOMA) scheme in a full-duplex (FD) relaying cognitive radio network (CRN), where a primary user (PU) communicates with the base station under the assistance of a secondary user (SU), in order to sufficiently utilize both idle and underutilized spectrum resources. When the PU does not exist, the SU identifies this idle licensed spectrum band by spectrum sensing and occupies it so that the waste of spectrum hole resource is avoided. When the PU exists, it allows the SU to share its spectrum band by NOMA to further increase the spectrum efficiency. By adopting FD mode, the proposed FD CNOMA scheme not only achieves better system performance than half-duplex mode by receiving and transmitting simultaneously, but also overcomes inherent issues of conventional CRNs. To characterize the performance of the proposed scheme, expressions of exact and asymptotic ergodic rates and system throughput are worked out. Accordingly, the high signal-to-noise ratio slopes for these ergodic rates are further derived. Simulation results verify the correctness of all these derivation results. They also illustrate that FD CNOMA outperforms other cooperative benchmarks in real-world scenarios in terms of both ergodic rates and system throughput.