Repurposing existing drugs to treat new diseases is a cost-effective alternative to de novo drug development, but there are millions of potential drug-disease combinations to be considered with only a small fraction being viable. In silico predictions of drug-disease associations can be invaluable for reducing the size of the search space. In this work we present a novel network of drugs and the diseases they treat, compiled using a combination of existing textual and machine-readable databases, natural-language processing tools, and hand curation, and analyze it using network-based link prediction methods to identify potential drug-disease combinations. We measure the efficacy of these methods using cross-validation tests and find that several methods, particularly those based on graph embedding and network model fitting, achieve impressive prediction performance, significantly better than previous approaches, with area under the ROC curve above 0.95 and average precision almost a thousand times better than chance.