An absolute measurement of the 235U fission cross section has been carried out using a 24NaBe photoneutron source with median neutron energy of 964 keV. A symmetric two-foil experiment was set up to measure the fission rate in a low-albedo laboratory, and variations in the source-to-foil spacing used to determine the room background. Fission fragments passing through a limited solid angle aperture were recorded from each foil by solid state tracketch techniques. The photoneutron source was calibrated after each run using the manganese bath method and the secondary national standard source NBS-II. A computed neutron source spectrum with 32 keV FWHM was derived by the Monte Carlo method and used in reducing the data to a cross section at 964 keV. The final value of 1.21 ± 0.025 barns is absolute in that, except for small corrections, its determination was independent of any other cross section data.