In the first X-ray astronomical observations in the early 60s, isotropic X-ray radiation was discovered - the cosmic X-ray background (CXB). According to the generally accepted paradigm, the CXB is composed of the radiation of numerous active galactic nuclei, and its broadband spectrum contains information about the growth history of supermassive black holes in the Universe. Despite the long history of X-ray studies and the rapid development of observational technology, the measurement of the exact characteristics of the CXB is associated with significant uncertainties. We measured the CXB spectrum in the energy range from 3 to 20 keV using the NuSTAR observations of selected extragalactic fields with a total exposure of 7 Ms. Our method uses spatial modulation of the CXB signal on the NuSTAR detectors through the telescope's side aperture. A record statistical measurement accuracy of less than one percent has been achieved, with an insignificant contribution of systematic noise. The observed differences in the intensity of the CXB in different directions on the sky are consistent with the expected dispersion of the CXB due to variations in the population of point sources, which opens up new possibilities for studying the angular fluctuations of the CXB on a scale of several square degrees.