Rosie Jones is third in the 20-player field at 10-under 134, a stroke better than first-round leader Michele Redman, who matched the course record on Thursday with a 65.
Driscoll was 18 shots behind leader Charley Hoffman, who will take a two-stroke advantage over 2012 U.S. Open champion Webb Simpson into Sunday's final round.
With five consecutive pars after that, Weekly finished at 14-under 266 for a one-stroke victory over Matt Kuchar, the second- and third-round leader who had a closing 68.