In this paper, Dyakonov surface waves (Dyakonov SWs) existing at the interface between a semi-infinite isotropic medium and a conductor-backed uniaxial slab are analyzed with the help of an exponential-matrix method. The boundary conditions at the interface are formulated using eigenvalues and eigenvectors of two partnering media. Based on this, the existence region of Dyakonov SWs is formulated and proven to be highly dependent on the thickness of the uniaxial slab. Some relevant characteristics of the propagating Dyakonov SWs, such as the distribution of the propagation constant, and the electric- and magnetic-field distributions, are introduced and investigated. In addition, this method can be applied to analyze other finite thickness structures.