As known from thermodynamic principles, the surface tension of a liquid decreases with increasing temperature. This property can be used to force a liquid film to climb a vertical substrate whose lower end is held warmer than the top. The vertical gradient in surface tension generates a surface shear stress that causes the liquid film to spread upward spontaneously in the direction of higher surface tension. Experimental investigations have shown that the application of a large temperature gradient produces a thin climbing film whose leading edge develops a pronounced capillary rim which breaks up into vertical rivulets. In contrast, smaller temperature gradients produce thicker films whose profiles decrease monotonically toward the substrate with no evidence of a rim or subsequent film breakup. We have previously shown within linear stability analysis that a climbing film can undergo a fingering instability at the leading edge when the him is sufficiently thin or the shear stress sufficiently large for gravitational effects to be negligible. In this work we show that thicker films which experience significant drainage cannot form a capillary rim and spread in stable fashion. Gravitational drainage helps promote a straight advancing front and complete surface coverage. Our numerical predictions for the entire shape and stability of the climbing him are in good agreement with extensive experiments published years ago by Ludviksson and Lightfoot (AIChE J. 17, 1166 (1971)). We propose that the presence of a counterflow which eliminates the capillary rim can provide a simple and general technique for stabilizing thermally driven films in other geometries.