French author Charles Perrault was inspired by Basile's story when he wrote "The Beauty Sleeping in the Wood" ("La Belle au Bois Dormant") in 1697. It was published in Tales and Stories of the Past With Morals, also known as Tales of Mother Goose. Many of these stories were then rewritten by the Brothers Grimm.
This is the story that is closer to what we are all familiar with. The prince awakens the sleeping princess, and then weds her and they have two children. His evil stepmother then tries to cook the children but is stopped and throws herself into a fire.