Nanopores--nanosized holes that can transport ions and molecules--are very promising devices for genomic screening, in particular DNA sequencing. Solid-state nanopores currently suffer from the drawback, however, that the channel constituting the pore is long, approximately 100 times the distance between two bases in a DNA molecule (0.5 nm for single-stranded DNA). This paper provides proof of concept that it is possible to realize and use ultrathin nanopores fabricated in graphene monolayers for single-molecule DNA translocation. The pores are obtained by placing a graphene flake over a microsize hole in a silicon nitride membrane and drilling a nanosize hole in the graphene using an electron beam. As individual DNA molecules translocate through the pore, characteristic temporary conductance changes are observed in the ionic current through the nanopore, setting the stage for future single-molecule genomic screening devices.