In this paper we consider the cooling of a flat sheet moving through a semi-infinite expanse of viscous fluid. The heat resistance of the sheet is assumed to be so small that the temperature can be considered uniform across the sheet. Precise conditions for this assumption to be valid are derived. The problem is solved first by means of a coordinate expansion, which can be proved to converge for all values of the expansion variable. Since this series cannot be used numerically downstream, an alternative series expansion, which applies downstream, is also derived. Special sections are devoted to deriving solutions valid for small or large values of the Prandtl number. Finally, expressions are obtained for the Nusselt number and the cooling length. It is found that cooling is determined by the smaller of two diffusivities, namely, the kinematic viscosity and the thermal diffusivity.