This paper presents a data-driven controller tuning method that includes a set of constraints for ensuring closed-loop stability. The approach requires a single experiment and can also be applied to nonminimum-phase or unstable systems. The tuning scheme uses an approximation of the closed-loop output error in the model-reference control problem. For linearly parameterized controllers, minimization of the correlation between this error and the reference signal leads to a convex optimization problem. A sufficient condition for closed-loop stability is introduced, which is implemented as a set of convex constraints on the Fourier transform of specific auto- and cross-correlation functions. As the data length tends to infinity, closed-loop stability is guaranteed. The quality of the estimated controller is analyzed for finite data length. The effectiveness of the proposed method is demonstrated in simulation as well as experimentally on a laboratory-scale mechanical setup.