In a 2009 interview with The Women on the Web, Leslie Stahl jokingly said that she sprays her hair with cement to keep every strand in place at all times. This suggests that her hair is her own, though the wig question did not come up directly.
Stahl explained further that while covering the White House for CBS, she needed to style her hair so that it did not fall. She needed to be camera-ready at a moment's notice, and that required keeping her hair and makeup as perfect as possible all day. At the time she used a hairspray that was actually named Cement. In the 2009 interview, she explained that she has been using a hairspray from Biolage that accomplishes the same thing.