We describe a sensitive method for measuring the stress birefringence of an optical window that utilizes a phase-measuring Fizeau interferometer incorporating a variable retarder and a nonpolarizing beam splitter. When we test a material in an interferometer cavity, the wave front transmitted through the material is deviated by the surfaces, inhomogeneity, and birefringence of the material. Birefringence causes the transmitted wave front to have different optical path difference (OPD) profiles for the vertical and horizontal orientations of linear polarization. Subtracting these OPD profiles reveals the amount of phase difference between the fast and slow axes of the material. Hence, birefringence may be calculated. Phase-measurement techniques and a computer-controlled interferometer employing a variable liquid-crystal retarder provide a fully automated instrument for measuring stress birefringence. The theoretical derivation, discussion of the instrument, and experimental results are presented.