  We present the two-loop corrected operator matrix elements contributing to
the scale evolution of the longitudinal spin structure function $g_1(x,Q^2)$
calculated up to finite terms which survive in the limit $\epsilon = N - 4 \to
0$. These terms are needed to renormalize the local operators up to third order
in the strong coupling constant $\alpha_s$. Further the expressions for the
two-loop corrected operator matrix elements can be inserted into one loop
graphs to obtain a part of the third order contributions to these matrix
elements. This work is a first step in obtaining the third order anomalous
dimensions so that a complete next-to-next-to-leading order (NNLO) analysis of
the above mentioned structure function can be carried out. In our calculation
particular attention is paid to the renormalization constant which is needed to
restore the Ward-identities violated by the HVBM prescription for the
$\gamma_5$-matrix in $N$-dimensional regularization.
