A method of preparing a library of fragments comprising sites of engineered nuclease-induced double stranded breaks in double-stranded DNA (dsDNA) molecules, the method comprising: providing dsDNA; randomly shearing the dsDNA to a defined average length; optionally end-repairing and/or A-tailing the sheared dsDNA; ligating a stem-loop adapter, comprising: a first region, preferably of about 12 nucleotides; a second region, preferably of about 5 nucleotides, that forms one or more loops and comprises a single deoxyuridine nucleotide adjacent to a palindromic sequence for intramolecular ligation; and a third region that is complementary to the first region with one additional nucleotide; contacting the library with enzymes that nick the dsDNA at the deoxyuridine and to remove a terminal 3' phosphate; incubating the nicked dsDNA under conditions sufficient to promote intramolecular ligation and formation of a sample comprising circular dsDNA molecules; contacting the sample with one or more exonucleases sufficient to degrade any DNA molecules that are not circular; treating the sample with an engineered nuclease to induce site-specific cleavage; optionally end-repairing and then A-tailing the resulting ends; ligating a sequencing adapter comprising: a first region of about 12 nucleotides; a second region of about 40 nucleotides that forms one or more hairpin loops and comprises a primer site compatible for use in PCR priming and/or sequencing; a third region of about 13 nucleotides that is complementary to the first region with one additional nucleotide; and a single deoxyuridine nucleotide between the second and third regions, to create a population wherein the DNA fragments that were cleaved by the nuclease have a sequencing adapter ligated to the ends; thereby preparing a library enriched for nuclease-cleaved adapter-ligated fragments. providing dsDNA; randomly shearing the dsDNA to a defined average length; optionally end-repairing and/or A-tailing the sheared dsDNA; ligating a stem-loop adapter, comprising: a first region, preferably of about 12 nucleotides; a second region, preferably of about 5 nucleotides, that forms one or more loops and comprises a single deoxyuridine nucleotide adjacent to a palindromic sequence for intramolecular ligation; and a third region that is complementary to the first region with one additional nucleotide; a first region, preferably of about 12 nucleotides; a second region, preferably of about 5 nucleotides, that forms one or more loops and comprises a single deoxyuridine nucleotide adjacent to a palindromic sequence for intramolecular ligation; and a third region that is complementary to the first region with one additional nucleotide; contacting the library with enzymes that nick the dsDNA at the deoxyuridine and to remove a terminal 3' phosphate; incubating the nicked dsDNA under conditions sufficient to promote intramolecular ligation and formation of a sample comprising circular dsDNA molecules; contacting the sample with one or more exonucleases sufficient to degrade any DNA molecules that are not circular; treating the sample with an engineered nuclease to induce site-specific cleavage; optionally end-repairing and then A-tailing the resulting ends; ligating a sequencing adapter comprising: a first region of about 12 nucleotides; a second region of about 40 nucleotides that forms one or more hairpin loops and comprises a primer site compatible for use in PCR priming and/or sequencing; a third region of about 13 nucleotides that is complementary to the first region with one additional nucleotide; and a single deoxyuridine nucleotide between the second and third regions, a first region of about 12 nucleotides; a second region of about 40 nucleotides that forms one or more hairpin loops and comprises a primer site compatible for use in PCR priming and/or sequencing; a third region of about 13 nucleotides that is complementary to the first region with one additional nucleotide; and a single deoxyuridine nucleotide between the second and third regions, to create a population wherein the DNA fragments that were cleaved by the nuclease have a sequencing adapter ligated to the ends; thereby preparing a library enriched for nuclease-cleaved adapter-ligated fragments.