State estimation is an integral component of autonomous Cyber-Physical Systems (CPSs) which require an estimate of the system state for their navigation and control logic. As the sensors used for state estimation may be prone to cyberattacks, they are typically accompanied by an on-board fault detection system which attempts to detect cyberattacks injected through the sensors. However, stealthy cyberattacks can remain undetected by the fault detection system and could cause catastrophic failure of the CPS. In this paper, we characterize the vulnerability of state estimation to such attacks and propose a cyberattack-resilient state estimation algorithm. The proposed algorithm is designed in conjunction with the fault detection system, and is able to mitigate the impact of stealthy cyberattacks and achieve robust state estimation performance across various system specifications and attack scenarios. We demonstrate this fact through theoretical analysis as well as using numerical simulations.