We report the first precise spectral measurement of fast neutrons produced in a deuterated plastic target irradiated by an ultraintense sub-picosecond laser pulse. The 500-fs, 50-J, 1054-nm laser pulse was focused on the deuterated polystyrene target with an intensity of 2 x 10(19) W/cm(2). The neutron spectra were observed at 55 degrees and 90 degrees to the rear target normal. The neutron emission was 7 x 10(4) per steradian for each detector. The observed neutron spectra prove the acceleration of deuterons and neutron production by d(d,n)3He reactions in the target. The neutron spectra were compared with Monte Carlo simulation results and the deuteron's directional anisotropy and energy spectrum were studied. We conclude that 2% of the laser energy was converted to deuterons, which has an energy range of 30 keV up to 3 MeV.