In this paper, a distributed model predictive control (DMPC) approach is proposed for the undirected networked vehicle platoon under state constraints. Unlike the existing results, Lyapunov and string stability of the closed-loop systems are simultaneously guaranteed for the undirected vehicle network. Specifically, the designed terminal constraints ensure the iterative feasibility of the DMPC algorithm. Then, the control design problem is formulated as a distributed optimization problem by using the state information of itself and its neighbors. The closed-loop stability of the system is established by using the cost functions as a Lyapunov function. In addition, two state constraints are designed to ensure the string stability under the constant distance strategy. Finally, the effectiveness of the proposed DMPC controller is verified by simulation results.