  The possibility is investigated to measure the polarized structure function
$g_1(x,Q^2)$ in the collider mode of HERA operating with a polarized lepton and
proton beam. The $x$ dependence of $g_1$ can be measured at a statistical
precision of $\sim 20\%~{\rm to}~80\%$ in the range $0.0005 < x < 0.5$
correlated to virtualities $15 < \langle Q^2 \rangle < 3500 \GeV^2$ at beam
polarizations $\lambda_p \sim \lambda_e = 0.8$ and ${\cal L}_{int} = 60
\PB^{-1}$. At a low energy option, $E_p = 300 \GeV$, the statistical accuracy
improves to values of $\sim 15\%~{\rm to}~50\%$ in the range $10^{-3} < x <
0.5$ and $10 < Q^2 < 800 \GeV^2$.
