Once you have a hint of interest, don’t approach her right away.  You want to confirm her interest.  The trick to doing so is to repeat the moment where your eyes meet.  This time you don’t want or need to be as subtle, but again, avoid staring.  Simply look for the moment where you can meet her eyes and, if they meet again, she’s showing you she’s interested in you. To be 100% certain, give her a nice, short smile.  If she returns it, you have a green light to approach her.
Approaching her
If you’ve already shared a moment of connection and a smile, you should approach her according to the situation.  If it’s a private place, you might want to wait until you’re both in a more public area to approach her.  If she’s in the middle of something, you might want to wait until she finishes what she’s doing.  The trick is to remember once contact is made, the interest is there and it won’t go away.  All you have to do is approach her before she goes away.

When you approach her, don’t launch upon her like a desperate clown.  Be slow and methodical, but confident.  Approach her with a smile and a single goal: to give her a compliment and your phone number.
Hand her your phone number with a flattery
When you first approach a girl you have never met before, the first thing you say and do, and how you hold yourself when you say and do it, can be the make or break of your chance to get with her.  This is why when you flirt for the first time you don’t want to hang around for too long such that you might say the wrong thing and blow it.

Just as making initial eye contact was about getting her interest, so too is your first approach.  When you first talk to her, you should excuse yourself by telling her that you hope she doesn’t mind you interrupting her but that she caught your interest – and you’d be pleased if she wouldn’t mind if you gave her your phone number.  Explain to her that it is entirely up to her if she calls you, and reinforce to her that there is no pressure and that you aren’t asking for her number.  Then tell her that you really hope she calls you because she is very attractive and you would like to get to know her better.

Don’t go crazy trying to compliment her.  A single flattering remark that you find her attractive is enough.  Over complimenting her will raise her defenses as she will begin wonder about your true motives.  Don’t give her anything to wonder about. Keep it short and simple. You want to flatter her and give her your phone number.  That is your first move.
A tip when making the first move
Girls are shy, at least, more shy than guys.  You’ll be amazed to learn that many gorgeous women are single because most guys think they must be taken or are out of their league.  Give her your number, but don’t ask for hers. Tell her you’d love to have her phone number and she might volunteer it, but most of all you don’t want to make her feel uncomfortable.  You’ve made the first move, and the second move is for her to call you.

When she calls
If you’ve followed the general guidelines given here you will have great success in attracting a girl’s interest.  She won’t know much about you, but she does know you’re friendly, you smile, you’re not overbearing and you didn’t try to be pushy in your desire to get to know her.  Above all else, she knows you think she’s attractive and she will also know there is no way she’ll see you again unless she calls you – because you didn’t ask for her number.  More often than not you’ll find she’ll call you that day or, at worst, within a few days.

When she calls, you will want to thank her and say that you are pleased she did.  Now that she has shown interest, you can chat with her if you want to, but it is recommended you keep the conversation short and simply ask her if she’d like to meet up (since she made the contact, meeting up for lunch, dinner or any other reason is fine, so long as you pick a public location).

Naturally it will be easy for you to ask her out since she called you.