Two-dimensional conjugate free convection due to a vertical plate of finite extent adjacent to a semi-infinite fluid region is investigated analytically and numerically. Computed solutions to the governing heat and momentum equations are obtained for a wide range of values of the non-dimensional parameters that are present in the problem, namely the Rayleigh number, Ra, the Prandtl number, Pr, the thermal conductivity ratio, k, between the plate and the fluid medium, and the plate aspect ratio, lambda. For Ra >> 1, the results give good agreement with an alternative formulation in which two-dimensional conduction in the solid is coupled with a convective boundary-layer flow in the fluid, the resulting non-linear system of equations then being solved iteratively. In addition, a third, much simpler, approach which assumes one-dimensional conduction in the plate produces accurate easily-obtained formulae for the average conjugate boundary temperature and Nusselt number.