For a bounded smooth domain $\Omega$ in the plane we consider the minimisation of the Willmore functional for graphs subject to Dirichlet boundary conditions. In a first step we show that sequences of functions with bounded Willmore energy satisfy uniform area and diameter bounds yielding compactness in $L^1(\Omega)$. We therefore introduce the $L^1$--lower semicontinuous relaxation and prove that it coincides with the Willmore functional on the subset of $H^2(\Omega)$ satisfying the given Dirichlet boundary conditions. Furthermore, we derive properties of functions having finite relaxed Willmore energy with special emphasis on the attainment of the boundary conditions. Finally we show that the relaxed Willmore functional has a minimum in $L^{\infty}(\Omega) \cap BV(\Omega)$. This is joint work with Hans--Christoph Grunau (Magdeburg) and Matthias Röger (Dortmund).