Many viruses use a powerful terminase motor to pump their genome inside an empty procapsid shell during virus maturation. The large terminase (TerL) protein contains both enzymatic activities necessary for packaging in such viruses: the ATPase that powers DNA translocation and an endonuclease that cleaves the concatemeric genome both at initiation and completion of genome packaging. However, how TerL binds DNA during translocation and cleavage is still mysterious. Here we investigate DNA binding and cleavage using TerL from the thermophilic phage P74-26. We report the structure of the P74-26 TerL nuclease domain, which allows us to model DNA binding in the nuclease active site. We screened a large panel of TerL variants for defects in binding and DNA cleavage, revealing that the ATPase domain is the primary site for DNA binding, and is required for nucleolysis. The nuclease domain is dispensable for DNA binding but residues lining the active site guide DNA for cleavage. Kinetic analysis of nucleolysis suggests flexible tethering of the nuclease domains during DNA cleavage. We propose that interactions with the procapsid shell during DNA translocation conformationally restrict the nuclease domain, inhibiting cleavage; TerL release from the procapsid upon completion of packaging unlocks the nuclease domains to cleave DNA.