I'm going to help you young men out here and give you something gold! Humor always works best.
Sit across from the girl and pass her written notes. In your note state, "I'm writing you this note not because I am obeying the library's rule of silence, but rather because I am a mute and can't talk. I can also read lips if you just want to mouth your response back to me.
Ask her some entertaining questions...
Then write something like, "it kind of sucks being a mute because I can't impress girls with my verbal wit, so I must do it in writing and it kind of sucks when there is no paper around. But I have enough paper here to carry a 10 hour conversation with you. SMILE!
I am every women's dream, "a man that can't speak allows more time for the women to talk!" then write, "I'm an excellent conversationalist on all the internet chat rooms and instant messaging...how about giving me your e-mail address so that we can communicate further."
Write crazy stuff like that, then when the bell rings tell her you miraculously got your voice back and would now prefer her phone number!