Inversion of the G loop of bacteriophage Mu requires the phage-encoded Gin protein and a host factor. The topological changes in a supercoiled DNA substrate generated by the two purified proteins were analyzed. More than 99% of the inversion products were unknotted rings. This result excludes synapsis by way of a random collision of recombination sites, because the resulting entrapped supercoils would be converted into knots by recombination. Instead, the recombination sites must come together in the synaptic complex in an ordered fashion with a fixed number of supercoils between the sites. The linking number of the substrate DNA increases by four during recombination. Thus, in three successive rounds of inversion, the change in linking number was +4, +8, and +12, respectively. These results lead to a quantitative model for the mechanism of Gin recombination that includes the distribution of supercoils in the synaptic complex, their alteration by strand exchange, and specific roles for the two proteins needed for recombination.