The neutron capture cross section of (235)U was measured for the neutron incident energy region between 4 eV and 1 MeV at the DANCE facility at the Los Alamos Neutron Science Center with an unprecedented accuracy of 2-3% at 1 keV. The new methodology combined three independent measurements. In the main experiment, a thick actinide sample was used to determine neutron capture and neutron-induced fission rates simultaneously. In the second measurement, a fission tagging detector was used with a thin actinide sample and detailed characteristics of the prompt-fission gamma rays were obtained. In the third measurement, the neutron scattering background was characterized using a sample of (208)Pb. The relative capture cross section was obtained from the experiment with the thick (235)U sample using a ratio method after the subtraction of the fission and neutron scattering backgrounds. Our result indicates errors that are as large as 30% in the 0.5-2.5 keV region, in the current knowledge of neutron capture as embodied in major nuclear data evaluations. Future modifications of these databases using the improved precision data given herein will have significant impacts in neutronics calculations for a variety of nuclear technologies.