  We calculate the $O(\alpha_s)$ corrections to longitudinal spin-spin
correlations in $e^+e^-\to q\bar q$. For top quark pair production the
$O(\alpha_s)$ corrections to the longitudinal spin-spin asymmetry amount to
less than 1% in the $q^2$-range from above $t\bar t$-treshold up to
$\sqrt{q^2}= 1000 GeV$. In the $e^+e^-\to b\bar b$ case the $O(\alpha_s)$
corrections reduce the asymmetry value from its $m=0$ value of -1 to
approximately -0.96 for $q^2$-values around the Z-peak. This reduction can be
traced to finite anomalous contributions from residual mass effects which
survive the $m\to 0$ limit. We discuss the role of the anomalous contributions
and the pattern of how they contribute to spin-flip and no-flip terms.
