  In this paper we construct a complete injective holomorphic immersion
$\mathbb{C}\to\mathbb{C}^2$ whose image is dense in $\mathbb{C}^2$. The
analogous result is obtained for any closed complex submanifold $X\subset
\mathbb{C}^n$ for $n>1$ in place of $\mathbb{C}\subset\mathbb{C}^2$. We also
show that, if $X$ intersects the unit ball $\mathbb{B}^n$ of $\mathbb{C}^n$ and
$K$ is a connected compact subset of $X\cap\mathbb{B}^n$, then there is a Runge
domain $\Omega\subset X$ containing $K$ which admits a complete holomorphic
embedding $\Omega\to\mathbb{B}^n$ whose image is dense in $\mathbb{B}^n$.
