How to approach her, at work

Hi there, I am writing from Paris France,
Got a question on how to approach this girl at work.
She basically comes to have lunch in my company restaurant every day, a 12:00.
She does not work here, she's from another company.
I noticed her, and I think she did notice me, few eyes contacts and smiles.
I iniated a "bonjour" :-), hi, and she responded well to it, so basically everyday we have our daily "Bonjour" when we meet. I call her the "bonjour girl".
Well..That's it!!! it's been like that for almost a month, many co-workers told me that there was something, but I just did'nt do anything, and don't know what to do, how to get to the next level.
One of the reasons of my non-action, except the fact that I realized that I was a bit shy, usualy I am not, it's the face that she is never alone, always with her co-workers, and so, not easy to talk to her.
Also talking to her at work, not easy with my co-workers around...
One last thing, I already have a girlfriend...I guess it's blocking me too .... so if you have any idea...
Almost forgot...as I am quite good with computers, I managed to find her on facebook!! I wanted to write to her on facebook, but I am scared that she thinks I am this weirdow who has been looking for her on the net...
thanks for your help in advance!
Au revoir!! :-)

finding people on FB is no longer considered 'stalkerish' especially if you're "known" to her for about a month. Just be quaint and to the point and say you've been admiring her from afar but thought you might be overstepping your 'professional' relationship talking to her at work or in front of her collegues. Tell her you'd liek to get to know her and ask her to something not super date like, like a coffee somewhere OTHER than where you work. Inviting her for a game of pool or darts is pretty low-pressure to, just be fun with it, I bet she'll reply and you can build there.

At work I used to say "I'm probably not supposed to be talking to girls at work but I noticed some non-sexual chemistry (they always laugh at that if said with a little grin, or emphasis on the "NON-sexual" because its kinda a disclaimer or easy way out, while still bringing it up..) and think you seem like a nice person I'd like to get to know." might tell her you notice she 'coincidently' keeps coming to the palce YOU work and ask if she has anything to do with the selecting of where they'll be dining. Keep it short, and even if its a no go you'll have somewhere to start without losing face at work!