Glassy 1-butene (CH2=CHCH2CH3) was studied by incoherent inelastic neutron scattering below 10 meV in the temperature range 18 less than or equal to T less than or equal to 80 K covering the glass transition temperature (T-g = 60 K). A broad excitation peak due to the low-energy excitation was found at 3-4 meV. The spectra corrected for the Bose factor could be scaled to a unique curve for temperatures below T-g. The absolute density of vibrational states was derived from the spectrum at 18 K and the previously measured heat capacity data. The number of vibrational states associated with the low energy excitation was 1.4 per molecule. This is much larger than those for SiO2 and other amorphous materials studied so far. The density of vibrational states was well reproduced by the soft potential model originally conceived for network glasses.