  Results are reported from the HERMES experiment at HERA on a measurement of
the neutron spin structure function $g_1^n(x,Q^2)$ 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 $\int_{0.023}^{0.6}
g_1^n(x) dx$ evaluated at a fixed $Q^2$ of $2.5 (GeV/c)^2$ is $-0.034\pm
0.013(stat.)\pm 0.005(syst.)$. Assuming Regge behavior at low $x$, the first
moment $\Gamma_1^n=\int_0^1 g_1^n(x) dx$ is $-0.037\pm 0.013(stat.)\pm
0.005(syst.)\pm 0.006(extrapol.)$.
