As the complexity of Cyber-Physical Systems (CPS) increases, it becomes more and more challenging to ensure the reliability of CPS, especially in the presence of system failures. Simplex architecture is shown to be an efficient tool to address the software failure in such systems. However, when physical failures also appear, Simplex does not work any more because the physical dynamics change due to physical failures. The Simplex architecture designed for the original physical model may not be suitable for the new dynamics. To address both software and physical failures, this paper presents the L1Simplex architecture, which contains the safety monitor, the high-performance controller (HPC), the L1-based high-assurance controller (HAC), and the decision logic for controller switching. The safety monitor is used to monitor the system behavior. It leads to another controller switching rule besides the stability-envelope-based rule in the decision logic. The HAC is designed based on the L1 adaptive controller, with which the stability envelope is computed. We show that the L1Simplex architecture can efficiently handle a class of software and physical failures.