The on-going EU sovereign debt crisis is causing great concern about the sustainability of national debt issued by the member states. In this paper, we propose a methodology to estimate the likelihood of the default of one or more countries in the Euro Area by extending the approach in Pianeti et al. (2012) to the case of multiple defaults. We provide an assessment of the marginal, the joint and the conditional default probabilities within the Euro Zone. The adopted measure of systemic risk is the probability of a joint default of the EU countries over a 5 years' time horizon. We find evidence of increasing systemic risk and danger of contagion from early-2007 and more significantly from late 2011 onwards. We show the forecasting capability of the estimated default probabilities through a comparison with the dynamics of a benchmark stock market index which marked the timeline of the recent sovereign debt crisis.