A new Internet protocol (IPv6) is being deployed with new features and improvements for the devices that are connected to the network. Unfortunately, the current Internet Protocol (IPv4) and the new Internet Protocol (IPv6) are incompatible with each other. During the time of migration from IPv4 to IPv6 networks, when both versions are available and the users of Internet want to connect without any restrictions, a transition mechanism is required. A number of transition mechanisms have been proposed by the Internet Engineering Task Force (IETF) in order to ensure smooth and independent changeover from IPv4 to IPv6 networks. In this paper, we propose a new flexible transition scheme which is known as the Bi-Directional Mapping System (BDMS) that deals with the IPv4/IPv6 address mapping transition as well as converting the received packets to their destination environment. Our new transition scheme (BDMS) is simple, efficient, and easy to implement. BDMS reduces the packet size compared with tunneling mechanisms and reduces the costs of the IPv6 network compared with the dual stack mechanisms.