Gather ideas for a simple date and choose the least complicated option. Dates don't have to be elaborate; they're about getting to know one another, not about engaging in an elaborate activity. Plan an activity that will give the two of you the opportunity to talk, such as a lunch or a coffee -- the simpler it is since fewer things can go wrong. Also, if you have a specific time and place in mind it will be easier to ask her on a date.
Approach your potential date when she's alone so that she doesn't feel on-the-spot in front of people. She is also more likely to make her own choice on her own, rather than feeling swayed by peer pressure in front of friends.
Ask her out in a decisive manner. Instead of "Do you want to do something sometime?" ask "Do you want to go to the park with me on Tuesday afternoon to feed the ducks?" This makes you seem more confident and collected, which will make you more attractive to her. If she says yes, firm up the plans by suggesting a time and a meeting place. If you can't finalize the plans right now, ask her for her phone number so you can call later.
Walk away with your head held high if she says no. Not everybody is going to be interested in you, and there are any number of reasons she may not feel up to dating right now. So, if she says no, don't try to convince her. Rather, just walk away. There will be more beautiful women.