A new exact solution is found for the equations of motion of a viscous gas for a stationary shear flow of hot (800–1500 K) gas between two parallel plates moving at different speeds (an analog of the incompressible Couette flow). One of the plates was considered thermally insulated. For the dependence of the coefficient of viscosity on temperature, the Sutherland formula is adopted. Unlike other known exact solutions, instead of a linear association between the viscosity and thermal conductivity coefficients, a more accurate formula was used to calculate the thermal conductivity coefficient, having the same accuracy in the temperature range under consideration as the Sutherland formula (2 %). Using the obtained exact solution, the qualitative effect of compressibility on the friction stress and the temperature, and velocity profiles were investigated. It is shown that the compressibility of the gas leads to an increase in the friction stress, if one of the plates is thermally insulated. The new exact solution was compared with the known exact solution (Golubkin, V.N. & Sizykh, G.B., 2018) obtained using the Sutherland formula for the viscosity coefficient and the Reynolds analogy for the thermal conductivity coefficient. It was found that both solutions lead to the same conclusions about the qualitative effect of compressibility on the friction stress and on the temperature and velocity profiles. However, the increase in friction stress caused by compressibility of the gas turned out to be underestimated twice when using the Reynolds analogy. This shows that the assumption of a linear relationship between the coefficients of viscosity and thermal conductivity can lead to noticeable quantitative errors.