Measurements of electric birefringence with increased sensitivity are possible using lasers with high intensity and stability, provided that perturbations resulting from stray light and strain in cell windows can be reduced. A new type of cell window is designed for minimal strain and is used in a standard birefringence setup with optimized components. The new instrument is characterized by a stray-light constant of 2 x 10(-7) and a negligible residual birefringence. Thus, measurements can be extended to small angles from the crossed position providing birefringence signals of high amplitudes at favorable signal-to-noise ratios. Special effects at small angles from the crossed position like a divergent increase of relative amplitudes to extreme values, a nonlinear response, a new type of electro-optical anomaly, and a simple bypass around this anomaly are observed and shown to be consistent with the theory. The technique proves to be particularly useful for measurements at physiological salt concentrations, where signals for most systems are too small under conventional conditions.