This realization happens to Bella in stages, actually. Bella is instantly drawn to Edward and singles him out as being different from the average guy, but it is Jacob who plants the seeds of thought to consider Edward as a vampire. In Chapter 6, Jacob passes off his information as slightly romanticized old tales, but the detail he conveys captivates Bella's attention. He tells her,

There's always a risk for humans to be around the cold ones, even if they're civilized like this clan was. You never know when they might get too hungry to resist . . . They claimed that they didn't hunt humans.

He goes on to explain to Bella that Edward's family belongs to this same clan of "cold ones." Bella tells him that he's a good storyteller, but then she goes home to do some internet research.

In Chapter 7, she finds a particularly striking entry online about the history of vampires:

Stregoni benefici: An Italian vampire, said to be on the side of goodness, and a mortal enemy of all evil empires.

She begins cataloging all that she reads online with her observations about Edward: "speed, strength, beauty, pale skin, eyes that shift color."

When Edward saves her life one more time in chapter eight, Bella becomes convinced that Edward can read minds. She finally decides to confront him with her body of knowledge in chapter nine, telling him, "It doesn't matter to me what you are."

Edward's voice becomes mocking, and he asks her, "You don't care if I'm a monster? If I'm not human?"

Bella replies with one simple word: "No."

It is at this point that Bella's convictions are clear. She is positive that Edward is a vampire, but she doesn't care because she has already fallen in love with him.

Bella comes to the conclusion that Edward is a vampire right around page 138-139. She realizes the truth but doesn't quite fully admit it to herself yet. When she realizes that he is a vampire, she thinks about two options: the first option is that she could tell him to leave her alone, run away from him, refuse to see him, and never talk to him again. The second option is to continue to act as she had been, to see him, to talk to him, to be with him, and accept him. She thinks that if he were going to hurt her, he would have done it already, and she believes that he is good. She makes her decision rather easily, because she is already falling in love with him. She really can't stay away. 

When she tells him she knows, she explains how she heard the story from Jacob Black, and he freezes, only barely hiding his nervousness. She tells him, "I decided it didn't matter." This happens in chapter 9, on page 184 of my edition. When this happens, the truth is out in the open, and she makes it clear to Edward that she trusts him and is willing to work things out to be near him, even though he believes this is wrong. Neither of them can bring themselves to stay away from each other.