A single-molecule method for sequencing DNA that does not require fluorescent labelling could reduce costs and increase sequencing speeds. An exonuclease enzyme might be used to cleave individual nucleotide molecules from the DNA, and when coupled to an appropriate detection system, these nucleotides could be identified in the correct order. Here, we show that a protein nanopore with a covalently attached adapter molecule can continuously identify unlabelled nucleoside 5'-monophosphate molecules with accuracies averaging 99.8%. Methylated cytosine can also be distinguished from the four standard DNA bases: guanine, adenine, thymine and cytosine. The operating conditions are compatible with the exonuclease, and the kinetic data show that the nucleotides have a high probability of translocation through the nanopore and, therefore, of not being registered twice. This highly accurate tool is suitable for integration into a system for sequencing nucleic acids and for analysing epigenetic modifications.