Want to ask her to be my girlfriend

So, I met this girl and we decided to date exclusively. We met on an online dating site and have just been meeting offline. We slept together and we both have intense emotions for each other. She deleted her account on the dating site and keeps saying stuff about me on her facebook profile that she likes me. Her last entries were "I heart Chris." "I had an amazing weekend with Chris, *sigh*" "Deleted my okcupid account, take that anyway you want =)"

We haven't been dating each other for a while, but I feel that I really like her and want her to be my girlfriend. I usually don't have sex with girls I date. She babysits and she's really good with kids and I just think she is a good person to have a relationship with. I've been on 2 dates with her (but you have to understand that we've been talking on the phone a lot before we met in person), but I think she's been hinting that she wants to be my girlfriend and I feel really strongly about her. Her friends know about me and she's just been giving off strong vibes and I just feel this urge to ask her to be my girlfriend.

Should I ask her to be my girlfriend? If so, how should I go about it?

Isn't the "exclusive dating" = relationship? OK, guess not since you're still asking.

I say ASK! Just grab her hand one day squeeze it and say "I'm going to introduce you to X as my girlfriend what do you think?" (X being any person you get to introduce her to). Given what you say here... your answer will be a huge beaming smile with YES slapped all over it.

Update: She said she wants to be my girl friend. She had a big smile on her face and was giggling happy.

Told ya. How did you ask?

(Insert Small Talk)

Me: So, where are we with our relationship/dating thing?

Her: Idk...

Me: Do you want to be my girl friend?

Her: Ya! (insert giggling here)

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I was just completely random about it.

Oh okay. well congrats.