Learning algorithms have enjoyed numerous successes in robotic control tasks. In problems with time-varying dynamics, online learning methods have also proved to be a powerful tool for automatically tracking and/or adapting to the changing circumstances. However, for safety-critical applications such as airplane flight, the adoption of these algorithms has been significantly hampered by their lack of safety, such as “stability,” guarantees. Rather than trying to show difficult, a priori, stability guarantees for specific learning methods, in this paper we propose a method for “monitoring” the controllers suggested by the learning algorithm online, and rejecting controllers leading to instability. We prove that even if an arbitrary online learning method is used with our algorithm to control a linear dynamical system, the resulting system is stable.