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We report the formation of solid-state nanopores using a scanning helium ion microscope. The fabrication process offers the advantage of high sample throughput along with fine control over nanopore dimensions, producing single pores with diameters below 4 nm. Electronic noise associated with ion transport through the resultant pores is found to be comparable with levels measured on devices made with the established technique of transmission electron microscope milling. We demonstrate the utility of our nanopores for biomolecular analysis by measuring the passage of double-strand DNA.