  We prove existence of solutions for the Benjamin-Ono equation with data in
$H^s(\R)$, $s>0$. Thanks to conservation laws, this yields global solutions for
$H^\frac 1 2(\R)$ data, which is the natural ``finite energy'' class. Moreover,
inconditional uniqueness is obtained in $L^\infty_t(H^\frac 1 2(\R))$, which
includes weak solutions, while for $s>\frac 3 {20}$, uniqueness holds in a
natural space which includes the obtained solutions.
