Results are reported from the HERMES experiment at HERA on a measurement of the neutron spin structure function g n 1 (x, Q 2 ) in deep inelastic scattering using 27.5 GeV longitudinally polarized positrons incident on a polarized 3 He internal gas target. The data cover the kinematic range 0.023 < x < 0.6 and 1 (GeV/c) 2 < Q 2 < 15 (GeV/c) 2 . The integral ∫0.0230.6 g n 1 (x) dx evaluated at a fixed Q 2 of 2.5 (GeV/c) 2 is -0.034 ± 0.013 (stat.)±0.005(syst.). Assuming Regge behavior at low x, the first moment Γ n 1 = ∫01 g n 1 (x) dx is -0.037±0.013(stat.)±0.005(syst.)±0.006(extrapol.).