We report on a precision measurement of the neutron spin structure function using deep inelastic scattering of polarized electrons by polarized ^3He. For the kinematic range 0.014<x<0.7 and 1 (GeV/c)^2< Q^2< 17 (GeV/c)^2, we obtain at an average . We find relatively large negative values for at low . The results call into question the usual Regge theory method for extrapolating to x=0 to find the full neutron integral , needed for testing quark-parton model and QCD sum rules.