  We show that the classical Brezis-Nirenberg problem $$ -\Delta u=u|u| +
\lambda u\ \hbox{in}\ \Omega,
  u=0\ \hbox{on}\ \partial\Omega,
  $$ when $\Omega$ is a bounded domain in $\mathbb R^6$ has a sign-changing
solution which blows-up at a point in $\Omega$ as $\lambda$ approaches a
suitable value $\lambda_0>0.$
