Previous works established the capacity region for some special cases of discrete memoryless degraded cognitive interference channel (CIC) with unidirectional destination cooperation (UDC). In this letter, we characterize the capacity region of the general discrete memoryless degraded CIC-UDC. The obtained results imply that the capacity region is achieved by the Gel'fand-Pinsker coding at the cognitive transmitter, superposition coding at the primary transmitter and decode-and-forward at the relay. Furthermore, using this general result and a novel converse analysis, we establish the capacity of the Gaussian degraded CIC-UDC, which had been open until this work.