For more than a decade, the strongest AIDS drugs could not fully control Matt Chappell's HIV infection. Now his body controls it by itself, and researchers are trying to perfect the gene editing that made this possible.

Scientists removed some of his blood cells, disabled a gene to help them resist HIV, and returned these "edited" cells to him in 2014. So far, it has given the San Francisco man the next best thing to a cure.

"I've been off medications for three and a half years," he said. He even was able to keep the virus in check despite cancer treatments last year that taxed his immune syst...