In this letter, a reconfigurable intelligent surface (RIS) is employed to enhance the secure performance of non-orthogonal unicast-multicast (NOUM) transmissions, where the propagation environment can be manipulated to achieve communication security while maintaining the quality of service (QoS) requirements of the NOUM system. Our objective is to maximize the secrecy rate of the unicast users under the QoS constraints on the multicast transmissions, where the multicast users are eavesdropping on the unicast messages. The joint optimization of the non-orthogonal power splitting and the RIS reflection matrix leads to a non-convex problem, to which two alternating optimization (AO) based solutions are developed based on the semi-definite programming (SDP) and penalty-based successive convex approximation (SCA). Our results demonstrate that secure transmission can be guaranteed in the RIS-aided NOUM system. Furthermore, the developed efficient solutions together with the RIS significantly improve the achievable security.