1. A method of modifying a genomic sequence in a cell, the method comprising introducing a polynucleotide encoding an artificial nuclease into the cell, the artificial nuclease comprising (i) first and second cleavage domains from an endonuclease, wherein the first cleavage domain is catalytically inactive and the second cleavage domain is catalytically active; and (ii) a DNA-binding molecule that is heterologous to the first and second cleavage domains, and further wherein the DNA-binding molecule of the nuclease binds to a target sequence in a double-stranded genome and the nuclease induces a site-specific single-stranded break at or near the target sequence in the double-stranded genome such that the genomic sequence is modified.