This work demonstrates the effectiveness of a novel simultaneous transmission and reflection reconfigurable intelligent surface (STAR-RIS) in Full-Duplex (FD) aided communication system. The objective is to minimize the total transmit power by jointly designing the transmit power and the transmitting and reflecting (T&R) coefficients of the STAR-RIS. To solve the nonconvex problem, an efficient algorithm is proposed by utilizing the alternating optimization framework to iteratively optimize variables. Specifically, in each iteration, we drive the closed-form expression for the optimal power design. The successive convex approximation (SCA) method and semidefinite program (SDP) are used to solve the passive beamforming optimization problem. Numerical results verify the convergence and effectiveness of the proposed algorithm, and further reveal in which scenarios STAR-RIS assisted FD communication defeats the Half-Duplex and conventional RIS.