Two broad classes of adaptive control algorithms can be found in the literature: i) stability based, with minimal assumptions on the plant; ii) performance based, with relatively more stringent assumptions on the plant. This paper proposes a solution, referred to as stability overlay (SO), to enable stability guarantees in performance based algorithms. In our methodology, the performance based adaptive control laws are only responsible for designating the controller that should be selected; the SO decides whether this controller should or not be used, based upon its most recent history of utilization. We argue that using two algorithms in parallel - the SO for stability purposes and any other suitable for the performance requirements - leads to higher levels of performance while guaranteeing stability of the adaptive closed-loop for bounded (but unknown) disturbances. The SO methodology is applicable to both time-invariant and time-varying, nonlinear and linear systems. However, due to space limitations, we only consider linear time-invariant (LTI) plants in this paper. The theory is illustrated with an example.