The neutron microscopic capture cross section for 235U is a critical parameter for the design and operation of nuclear reactors. The evaluated nuclear data libraries of ENDF/B-VII.1 and JENDL-4.0 have nearly identical values for the neutron capture cross section for neutron energies below 0.5 keV. In the most recent release of the JENDL library the onset of the unresolved resonance region was changed from 2.25 keV to 0.5 keV. In the energy region from 1.5 keV to 2.25 keV the average neutron capture cross section from ENDF/B-VII.1 is about 10% higher than that from JENDL-4.0. In an attempt to address the discrepancies between the libraries, a measurement of the neutron capture cross section of 235U was conducted at the Gaerttner LINAC Center located at Rensselaer Polytechnic Institute. This measurement used a 16-segment -multiplicity NaI(Tl) detector to detect the prompt gammas emitted from neutron interactions with a highly enriched 235U sample. Using the time-of-flight method, detected events were recorded and grouped based on the total gamma energy per interaction and observed multiplicity. A method was developed to separate fission from capture based on total energy deposition and gamma multiplicity. Application of this method in the thermal and resonance region below 0.5 keV for both the fission and capture produced cross sections that are in good agreement with both ENDF/B-VII.1 and JENDL-4.0 evaluations. The measurements support a lower 235U neutron capture cross section in the energy range 0.5 to 2.25 keV, which is closer to JENDL 4.0.