An analysis is presented of the steady two-dimensional spread of a flame down a vertical surface of a thermally thick solid fuel, under the assumptions that the gas-phase reaction is infinitely fast and that vaporization of the solid occurs at a constant temperature. A condition of regularity of the solid phase temperature, the gas phase mixture fraction and enthalpy, and the mass vaporization flux at the vaporization front is proposed to determine the flame spread rate. Numerical solutions computed using this condition show that the spread rate decreases with decreasing heat of reaction and becomes zero for a certain heat of reaction. An order-of-magnitude analysis is given for the case when heat conduction in the solid dominates heat conduction in the gas.