We use a network model of credit risk to measure market expectations of the potential spillovers from a sovereign default. Specifically, we develop an empirical model, based on the recent theoretical literature on contagion in financial networks, and estimate it with data on sovereign credit default swap spreads and the detailed structure of financial linkages among thirteen European sovereigns from 2005 to 2011. Simulations from the estimated model show that a sovereign default generates only small spillovers to other sovereigns. These results imply that credit markets do not demand a significant premium for the interconnectedness of sovereign debt in Europe.