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We study how the class of observers introduced in a previous paper, here referred to as circle-criterion observers, can be incorporated in output-feedback control. Due to the absence of a controller-observer separation property for nonlinear systems, the certainty-equivalence implementation of a state-feedback design may lead to severe forms of instability. We show, on the contrary, that the state-dependent convergence properties of circle-criterion observers can prevent such instabilities. Exploiting these convergence properties we develop a modified circle-criterion observer design that guarantees global asymptotic stability for certainty-equivalence controllers.