  For the two-phase membrane problem $ \Delta u = {\lambda_+\over 2}
\chi_{\{u>0\}} - {\lambda_-\over 2} \chi_{\{u<0\}} ,$ where $\lambda_+> 0$ and
$\lambda_->0 ,$ we prove in two dimensions that the free boundary is in a
neighborhood of each ``branch point'' the union of two $C^1$-graphs. We also
obtain a stability result with respect to perturbations of the boundary data.
Our analysis uses an intersection-comparison approach based on the Aleksandrov
reflection. In higher dimensions we show that the free boundary has finite
$(n-1)$-dimensional Hausdorff measure.
