This paper presents a data-driven controller-tuning algorithm that includes a sufficient condition for closed-loop stability. This stability condition is defined by a set of convex constraints on the Fourier transform of specific auto- and cross-correlation functions. The constraints are included in a correlation-based controller-tuning method that solves a model-reference problem. This entirely data-driven method requires a single experiment and can also be applied to nonminimum-phase and unstable systems. The resulting controller is guaranteed to stabilize the plant as the data length tends to infinity. The performance with finite data length is illustrated through a simulation example.