  Given a closed complex hypersurface $Z\subset \mathbb{C}^{N+1}$
$(N\in\mathbb{N})$ and a compact subset $K\subset Z$, we prove the existence of
a pseudoconvex Runge domain $D$ in $Z$ such that $K\subset D$ and there is a
complete proper holomorphic embedding from $D$ into the unit ball of
$\mathbb{C}^{N+1}$. For $N=1$, we derive the existence of complete properly
embedded complex curves in the unit ball of $\mathbb{C}^2$, with arbitrarily
prescribed finite topology. In particular, there exist complete proper
holomorphic embeddings of the unit disc $\mathbb{D}\subset \mathbb{C}$ into the
unit ball of $\mathbb{C}^2$.
  These are the first known examples of complete bounded embedded complex
hypersurfaces in $\mathbb{C}^{N+1}$ with any control on the topology.
