  Kummer's conjecture states that the relative class number of the $p$-th
cyclotomic field follows a strict asymptotic law. Granville has shown it
unlikely to be true -- it cannot be true if we assume the truth of two other
widely believed conjectures. We establish a new bound for the error term in
Kummer's conjecture, and more precisely we prove that
$\log(h_p^-)=\frac{p+3}{4}\log(p) +
\frac{p}{2}\log(2\pi)+\log(1-\beta)+O(\log_2 p)$, where $\beta$ is a possible
Siegel zero of an $L(s,\chi)$, $\chi$ odd.
