A major problem of the directed deletion of double-stranded DNA using Exonuclease III (Chapter 8 ) is the necessity to find two enzyme sites to open up the insert for digestion and to protect the primer site. The “Cyclone Sequencing” technique of Dale, McClare, and Houchins (1 ) overcomes this problem as it is totally independent of the restriction sites in the insert. In this procedure single-stranded Ml3 is hybridized to an oligomer that spans the Hin dIII (even-numbered M13 phage) or Eco RI site (odd-numbered phage) of the polylinker, generating a region of double-stranded DNA. This region is digested with the appropriate enzyme, linearizing the phage, and then the 3′ to 5′ exonuclease activity of T4 DNA polymerase is used to digest through the insert. The 3′ end is then homopolymer tailed using terminal transferase and the original oligomer used to create the double-stranded region, which has a 3′ end complementary to the homopolymer tail, is reannealed across the restriction site and the homopolymer tail. After ligation the DNA is used to transform competent cells and the resulting plaques picked at random for screening.