  We use the density-matrix renormalization group technique developed by White
\cite{white} to calculate the spin correlation functions
$<{S}_{n+l}^z{S}_n^z>=(-1)^l \omega(l,N)$ for isotropic Heisenberg rings up to
$N=70$ sites. The correlation functions for large $l$ and $N$ are found to obey
the scaling relation $\omega(l,N)=\omega(l,\infty)f_{XY}^{\alpha} (l/N)$
proposed by Kaplan et al. \cite{horsch} , which is used to determine
$\omega(l,\infty)$. The asymptotic correlation function $\omega(l,\infty)$ and
the magnetic structure factor $S(q=\pi)$ show logarithmic corrections
consistent with $\omega(l,\infty)\sim a\sqrt{\ln{cl}}/l$, where $c$ is related
to the cut-off dependent coupling constant $g_{eff}(l_0)=1/\ln(cl_0)$, as
predicted by field theoretical treatments.
