  We study the density of polynomials in $H^2(\Omega,e^{-\varphi})$, the space
of square integrable holomorphic functions in a bounded domain $\Omega$ in
$\mathbb{C}$, where $\varphi$ is a subharmonic function. In particular, we
prove that the density holds in Carath\'{e}odory domains for any subharmonic
function $\varphi$ in a neighborhood of $\overline{\Omega}$. In
non-Carath\'{e}odory domains, we prove that the density depends on the weight
function, giving examples.
