Planar Goubau lines show promise as high frequency, low-loss waveguides on a substrate. However, to date only numerical simulations and experimental measurements have been performed. This paper analytically investigates the surface wave mode propagating along a planar Goubau line consisting of a perfectly conducting circular wire on top of a dielectric substrate of finite thickness but infinite width. An approximate equation for the propagation constant is derived and solved through numerical integration. The dependence of the propagation constant on various system parameters is calculated and the results agree well with full numerical simulations. In addition, the spatial distribution of the longitudinal electric field is reported and excellent agreement with a numerical simulation and previous studies is found. Moreover, validation against experimental phase velocity measurements is also reported. Finally, insights gained from the model are considered for a Goubau line with a rectangular conductor. The analytic model reveals that the propagating mode of a planar Goubau line is hybrid in contrast to the transverse magnetic mode of a classic Goubau line.