  In this paper we investigate proper $\mbb{R}$--actions on hyperbolic Stein
surfaces and prove in particular the following result: Let $D\subset\mbb{C}^2$
be a simply-connected bounded domain of holomorphy which admits a proper
$\mbb{R}$--action by holomorphic transformations. The quotient $D/\mbb{Z}$ with
respect to the induced proper $\mbb{Z}$--action is a Stein manifold. A normal
form for the domain $D$ is deduced.
