The fission cross sections of 235U and 239Pu are measured with very high neutron energy resolution (0.17 ns/m) in the energy region from 100 to 2000 eV for 235U and to 20000 eV for 239Pu. The purpose of this measurement is to provide fission cross sections with energy resolution comparable with that available from transmission measurements for the purpose of deriving multilevel resolved resonance parameters. Fission ion chambers are used to detect fission fragments, and a 10B ionization chamber is used to measure the relative neutron flux at the 86-m flight path of the Oak Ridge Electron Linear Accelerator. The measured fission cross sections are the highest resolution measurements of good accuracy reported in the neutron energy range above 400 eV.