Present estimates suggest there are over 1 million virus species found in mammals alone, with about half a million posing a possible threat to human health. Although previous estimates assume linear scaling between host and virus diversity, we show that ecological network theory predicts a non-linear relationship, produced by patterns of host sharing among virus species. To account for host sharing, we fit a power law scaling relationship for host-virus species interaction networks. We estimate that there are about 40,000 virus species in mammals (including ~10,000 viruses with zoonotic potential), a reduction of two orders of magnitude from present projections of viral diversity. We expect that the increasing availability of host-virus association data will improve the precision of these estimates and their use in the sampling and surveillance of pathogens with pandemic potential. We suggest host sharing should be more widely included in macroecological approaches to estimating biodiversity.