  In this paper we prove that the Benjamin-Ono equation admits an analytic
Birkhoff normal form in an open neighborhood of zero in $H^{s}_{0}(\T, \R)$ for
any $s>-1/2$ where $H^{s}_{0}(\T, \R)$ denotes the subspace of the Sobolev
space $H^{s}(\T, \R)$ of elements with mean $0$. As an application we show that
for any $-1/2<s<0$, the flow map of the Benjamin-Ono equation $\mathcal{S}_0^t
: H^{s}_{0}(\T, \R)\to H^{s}_{0}(\T, \R)$ is nowhere locally uniformly
continuous in a neighborhood of zero in $H^{s}_{0}(\T, \R)$.
