You can cure cancer with diet, along with generally healthy living. "Cure" perhaps isn't the best word though, because bad diet (combined with other things like stress, lack of exercise, etc.) is frequently the cause of cancer in the first place.

EDIT: I didn't mean to say that changing to a healthy diet/lifestyle after being diagnosed will magically cure you (if that's why I'm being downvoted) but I believe it certainly can in some cases. If it's for lack of references, well, when it comes to cancer, you can find a study to support just about any view you want.