We describe semiparametric estimation and inference for causal effects using observational data from a single social network. Our asymptotic results are the first to allow for dependence of each observation on a growing number of other units as sample size increases. In addition, while previous methods have implicitly permitted only one of two possible sources of dependence among social network observations, we allow for both dependence due to transmission of information across network ties and for dependence due to latent similarities among nodes sharing ties. We propose new causal effects that are specifically of interest in social network settings, such as interventions on network ties and network structure. We use our methods to reanalyze an influential and controversial study that estimated causal peer effects of obesity using social network data from the Framingham Heart Study; after accounting for network structure we find no evidence for causal peer effects.