  Let $r_k(C_{2m+1})$ be the $k$-color Ramsey number of an odd cycle $C_{2m+1}$
of length $2m+1$. It is shown that for each fixed $m\ge2$,
\[r_k(C_{2m+1})<c^{k}\sqrt{k!}\] for all sufficiently large $k$, where
$c=c(m)>0$ is a constant.
  This improves an old result by Bondy and Erd\H{o}s (Ramsey numbers for cycles
in graphs, J. Combin. Theory Ser. B 14 (1973) 46-54).
