The neutron capture widths of the s-wave resonances at 13.9 and 33.8 keV in 64Ni have been determined using a setup with extremely low neutron sensitivity completely different from all previous experiments on this isotope. This feature is important because these resonances exhibit a very large scattering-to-capture ratio. A pulsed 3-MV Van de Graaff accelerator and a kinematically collimated neutron beam, produced via the 7Li(p, n) reaction, was used in the experiments. Capture gamma rays were observed by three Moxon-Rae detectors with a graphite, a bismuth-graphite, and a bismuth converter, respectively. The samples were positioned at a neutron flight path of only 6 to 8 cm. Thus, events due to capture of resonance-scattered neutrons in the detectors or in surrounding materials are completely discriminated by their additional time of flight. The short flight path and the high neutron flux at the sample position allowed for a signal-to-background ratio of approximately unity even for the broad resonance at 33.8 kev. The data obtained with the individual detectors were corrected for the efficiency of the different converter materials. For that purpose, detailed theoretical calculations of the capture gamma-ray spectra of the measured isotope and of gold, which was used as a standard, were performed. The final radiative widths are Γγ(13.9 kev) = 1.01 ± 0.07 eV and Γγ(33.8 kev) = 1.16 ± 0.08 Ev, considerably smaller than the rough estimates obtained in previous work.