Emerging infectious diseases are among the main threats to conservation of biological diversity. A crucial task facing epidemiologists is to predict the vulnerability of populations of endangered animals to disease outbreaks. In this context, the network structure of social interactions within animal populations may affect disease spreading. However, endangered animal populations are often small and to investigate the dynamics of small networks is a difficult task. Using network theory, we show that the social structure of an endangered population of mammal-eating killer whales is vulnerable to disease outbreaks. This feature was found to be a consequence of the combined effects of the topology and strength of social links among individuals. Our results uncover a serious challenge for conservation of the species and its ecosystem. In addition, this study shows that the network approach can be useful to study dynamical processes in very small networks.