We present a new quantitative x-ray phase-contrast imaging method based on the edge illumination principle, which allows achieving unprecedented nanoradian sensitivity. The extremely high angular resolution is demonstrated theoretically and through experimental images obtained at two different synchrotron radiation facilities. The results, achieved at both very high and very low x-ray energies, show that this highly sensitive technique can be efficiently exploited over a very broad range of experimental conditions. This method can open the way to new, previously inaccessible scientific applications in various fields including biology, medicine and materials science.