Accurate prediction of liquid–vapor phase change phenomena is critical in the design of thin vapor chambers and microheat pipes for the thermal management of miniaturized electronic systems. In view of this, we have considered the heat and mass transfer between two-liquid meniscuses separated by a thin gap of its own vapor. Assuming the heat and mass flow are to be steady and one-dimensional, analytic solutions are obtained to the linearized equations from the regularized 26-moment framework. Our analytic solutions provide excellent predictions for the effective heat conductivity of a dilute gas with those from the molecular dynamics (MD) and Boltzmann equation where Fourier's law fails. We also verified that the predicted heat and mass flow rates over the whole range of the Knudsen number are consistent with the kinetic theory of gases. Further, the model has been used to predict the effect of evaporation and accommodation coefficients on the heat and mass transfer between the liquid layers.