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This paper proposes a novel non-orthogonal multiple access (NOMA)-based cooperative transmission scheme for a spectrum-sharing cognitive radio network (CRN), whereby a secondary transmitter (ST) serves as a relay and helps transmit the primary and secondary messages simultaneously by employing NOMA signaling. This cooperation is particularly useful when the ST has good channel conditions to a primary receiver but lacks the radio spectrum. To evaluate the performance of the proposed scheme, the outage probability and system throughput for the primary and secondary networks are derived in closed forms. Simulation results demonstrate the superior performance gains for both networks thanks to the use of the proposed NOMA-based cooperative transmission scheme. It is also revealed that NOMA outperforms conventional orthogonal multiple access (OMA) and achieves better spectrum utilization.