  We prove that for a bounded simply connected domain $\Omega\subset \mathbb
R^2$, the Sobolev space $W^{1,\,\infty}(\Omega)$ is dense in
$W^{1,\,p}(\Omega)$ for any $1\le p<\infty$. Moreover, we show that if $\Omega$
is Jordan, then $C^{\infty}(\mathbb R^2)$ is dense in $W^{1,\,p}(\Omega)$ for
$1\le p<\infty$.
