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With the twofold cost of sex, derived asexual organisms have an immediate reproductive advantage over their sexual sisters. Yet the "twiggy" phylogenetic distribution of asexual lineages implies that they become extinct relatively quickly over evolutionary time. Meanwhile, bacteria and archaea have persisted for billions of years without requiring sexual reproduction. A simple explanation for this difference is that prokaryotes have very large population sizes that are not subject to the accumulation of deleterious mutations, but this implies that drift and mutational meltdown dominate derived asexual populations. Here, we explore a different hazard, quantifying the degree to which genetic variation is lost in asexual populations experiencing selective sweeps. Even though large populations generate diversity by mutation during sweeps, we find that populations that are safe from mutational meltdown may still be reduced to dangerous effective population sizes by sweeps. Thus, ironically, adaptation itself reduces further adaptive potential and may predispose asexual populations to extinction. We derive a simple approximation for the effective population size after a hard sweep and explore the impact of recent sweeps on evolutionary rescue. These factors may help to explain the phylogenetic twigginess of asexuals, the maintenance of sex and recombination, and the evolutionary persistence of prokaryotes. [ABSTRACT FROM AUTHOR]