  We prove that the initial value problem (IVP) for the BBM equation is
ill-posed for data in $H^s(\R)$, $s<0$ in the sense that the flow-map
$u_0\mapsto u(t)$ that associates to initial data $u_0$ the solution $u$ cannot
be continuous at the origin from $H^s(\R)$ to even $\mathcal{D}'(\R)$ at any
fixed $t>0$ small enough. This result is sharp.
