Gibson assembly and assembly-in-yeast are strategies to create long synthetic DNAs from diverse fragments, for example, when engineering bacteriophage genomes. Design for these methods requires terminal sequence overlaps in the fragments, determining the order of assembly. Design to rebuild a genomic fragment that is too long for a single PCR presents a puzzle since some candidate joint regions cannot yield satisfactory primers for the overlap. No existing overlap assembly design software is open-source, and none explicitly supports rebuilding. We describe here bigDNA software that solves the rebuilding puzzle by recursive backtracking, with options to remove or introduce genes; it also tests for mispriming on the template DNA. BigDNA was tested with 3082 prophages and other genomic islands (GIs), from 20 to 100 kb, and the synthetic Mycoplasma genitalium genome. Rebuilding assembly design succeeded for all but ∼1% of GIs. BigDNA will speed and standardize assembly design.