In New Moon, Edward calls her Isabella Marie Swan. Its near the end, when they are back from Italy and Bella is not sure about everything with him and kind of doubts him. He uses her full name, i guess to get her attention.
Her full name is Isabella Marie Swan. It says every character's full name on the New Moon website (newmoonthemovie.com) and that website is SO awesome! You just have to go to the Cullen House or the Quileute Reservation and if you click on the thing by the character then it tells all about them. :)
In the end of New Moon,
Bella said to Edward about him not loving her.
The quote:
"Isabella Marie Swan. You think that I went to the Volturi because I felt guilty?"
Yep. That's how I found out. :)
Edward also said Bella's full name during the proposal in Eclipse.