[Edward to Bella:] "Isabella." He pronounced my full name carefully, then playfully ruffled my hair with his free hand. A shock ran through my body at his casual touch. "Bella, I couldn't live with myself if I ever hurt you. You don't know how it's tortured me." He looked down, ashamed again. "The thought of you, still, white, cold…to never see you blush scarlet again, to never see that flash of intuition in your eyes when you were though my pretenses...it would be unendurable." He lifted his glorious, agonized eyes to mine. "You are the most important thing to me now. The most important thing to me ever." (13.125)
Here Edward declares his love. He's just told Bella about how conflicted he feels – how he does thirst for her blood as well as for her company. But here he seems to be making a vow never to hurt her, and never to let his vampire instincts get out of control. When he uses Bella's full given name, we know he's serious.