We introduce a new systemic risk measure, the change in conditional joint probability of default (Delta CoJPoD), that assesses the effects of interdependence within the financial system on the general financial system default risk. We apply our measure to examine the fragility of the European financial system during the ongoing sovereign debt crisis, encompassing 10 euro area sovereigns and 44 European Union banks in the period 01.01.2008 to 31.12.2011. Our results show that joint distress risk has increased since the end of 2009, parallel to decoupling of investors' perceptions about individual sovereign default risk. Overall, a default of Germany would have the highest contribution to systemic risk, while the effect of Greek default is limited. We use Spain as a default trigger to capture the effect of the sovereign debt crisis on the EU banking system. We find evidence for "too-big-to-save", riskiness-of-business and asset quality considerations when investors assess the banking system's vulnerability to sovereign risk. Leverage seems to be less informative in that respect. Our model could be an integral part of a policy makers' tool set to evaluate the usefulness and feasibility of bailout measures.