A gravity wave anelastic dispersion relation is derived that includes molecular viscosity and thermal diffusivity to explore the damping of high-frequency gravity waves in the thermosphere. The time dependence of the wave amplitudes and general ray trace equations are also derived. In the special case that the thermal structure is isothermal and the Prandtl number (Pr) equals 1, exact linear solutions are obtained. For high-frequency gravity waves with ω _Ir /N 1 an upward propagating gravity wave dissipates at an altitude given by z ₁ + H ln(ω _Ir /2H∣m∣³ν₁), where H is the density scale height, N is the buoyancy frequency, ν₁ is the viscosity at z = z ₁, and ω _Ir and m are the gravity wave intrinsic frequency and vertical wave number, respectively. Thus high-frequency gravity waves with large vertical wavelengths dissipate at the highest altitudes, resulting in momentum and energy inputs extending to very high altitudes. We find that the vertical wavelength of a gravity wave with an initially large vertical wavelength decreases significantly by the time it dissipates just below where it begins to reflect. The effect of diffusion on a gravity wave is similar to the effect of shear in the sense that as the molecular viscosity and thermal diffusivity increase due to decreasing background density, the intrinsic frequency plus mν/H decreases and the vertical wave number increases in order to satisfy the dispersion relation for Pr = 1. We also briefly explore the results with different Prandtl numbers using numerical ray tracing. Gravity waves in a Pr = 0.7 environment dissipate just a few kilometers below those in a Pr = 1 environment when H = 7 km, showing the utility of the analytic, Pr = 1 solutions.