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We find a new obstruction for a real Einstein 4-orbifold with anA1-singularity to be a limit of smooth Einstein 4-manifolds. The obstruction isa curvature condition at the singular point. For asymptotically hyperbolicmetrics, with boundary at infinity a conformal metric, we prove that if theobstruction vanishes, one can desingularize Einstein orbifolds with suchsingularities. The Dirichlet problem consists in finding Einstein metrics withgiven conformal infinity on the boundary: we prove that our obstruction definesa wall in the space of conformal metrics on the boundary, and that all theEinstein metrics must have their conformal infinity on one side of the wall.

D\'esingularisation de m\'etriques d'Einstein. I