An elastohydrodynamic solution for a high-pressure, low-clearance metal seal is presented. The fluid flow is assumed to satisfy Reynolds equation of hydrodynamic lubrication, and the deformation of the shaft and the seal is governed by the linear theory of elasticity. The viscosity of the fluid is assumed to have an exponential dependence on the pressure, while the density of the fluid is a linear function of the pressure. Closed-form solutions are obtained for two asymptotic limiting cases: (i) when the length of the seal is much greater than the radius of the shaft, and (ii) when it is much less. For intermediate ratios of the seal length to shaft radius, solutions are obtained numerically and examples are given to show the effect of seal length on the rate of mass flow.