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A newly developed neutron time-of-flight (nTOF) diagnostic with a fast instrument response function has been fielded on the OMEGA laser in a highly collimated line of sight. By using a small plastic scintillator volume, the detector provides a narrow instrument response of 1.7 ns full width at half maximum while maintaining a large signal-to-noise ratio for neutron yields between 10 10 and 10 14. The OMEGA hardware timing system is used along with an optical fiducial to provide an absolute nTOF measurement to an accuracy of ∼56 ps. The fast instrument response enables the accurate measurement of the primary deuterium-tritium neutron peak shape, while the optical fiducial allows for an absolute neutron energy measurement. The new detector measures the neutron mean energy with an uncertainty of ∼7 keV, corresponding to a hot-spot velocity projection uncertainty of ∼12 km/s. Evidence of bulk fluid motion in cryogenic targets is presented with measurements of the neutron energy spectrum.