A recent series of spectacular medical cases has electrified researchers around the world—cases in which therapy has allowed the HIV patient to permanently keep the virus under control. In 2009, doctors at Berlin’s Charité Hospital reported on a HIV-positive man named Timothy Ray Brown, known as the Berlin patient, who received a bone-marrow stem-cell transplant as treatment for leukemia. The donor was, thanks to genetics, immune to HIV—and the immunity seemed to have passed to Brown, who no longer needed antiretroviral therapy to control the HIV. Doctors eventually declared him cured. In 2013, doctors at Harvard Medical School reported similar...