Newman,
First, be truthful to yourself...is there any truth to the 'rumors' that she heard early on? Is this girl who called anything more than a friend, or better yet do you have feelings for this friend? If you do, girls can tell.
If there's not even a smidge of truth to the rumors and the friend is truly only that and you really, really like this girl then you need to work on her. Don't worry about the right thing or the wrong thing to say to 'get the girl', just be truthful and honest with her. (Don't do it in a letter, email or text)
Don't think she's as crazy as some people have deduced. She may have heard other rumors that she's trying to process. Good chance she's battling with what she hears versus what she feels. If what she's heard is true, she'll feel like the biggest fool for giving you a shot. If it's not and she doesn't give you a shot, she could regret it. My guess is that's her dilemma.
Put your heart on the line if you think she's worth it. If you truly believe she's the one, pursue it with truth. You've got nothing to lose. If she doesn't want to give it a try, wish her the best and move along.
And yes, I am speaking from experience. I heard rumors about a girl that I had started dating. At first I was very standoffish around her and I felt hurt and betrayed. Eventually, I confronted her with what I had heard, with details, and we discussed it in depth. We got to the bottom of it and what it came down to was jealous people saying hurtful things. If we had never talked it out, who knows how things would have turned out. I've been with that girl for 20 years now.
Best of luck to you.