Lend an ear. Get to know your prospective new date by befriending her -- and listening to her. Don't pry or be insensitive to her need for privacy by bombarding her with questions, as she may be experiencing residual pain from her breakup. However, in the course of your conversations, if she begins opening up about her last relationship, let her talk -- and listen attentively. The details may be difficult for you to hear, but it will let you know the status of her recovery process. Use what you learn to determine how long to wait before you ask her out.
Be sensitive and patient. Depending on the length and seriousness of her previous relationship, thinking of a new one may be uncomfortable and even painful for her. Give her time to process what she's going through, so you're not just a rebound date. If you have mutual friends in common, try to see her in a group situation several times before asking her out. If not, email, text or chat with her on the phone to learn more about each other before you ask her for a date. She'll get to know and trust you, as well as resist making assumptions about your motives, if you continue to listen and wait until she's ready to begin dating again.
Ask her out once you're confident that the time is. Plan the date ahead of time and approach her with a solid proposition -- not just a "Let's get together and go out sometime." Try to choose a date that will give you time alone to talk, but isn't too intimate a situation. For example, a movie that she mentioned she wants to see, followed by dinner at a quiet restaurant might be just right. You wouldn't want to ask her to go away with you for a weekend in the mountains as a first date because it's just too much too soon and could get awkward. If she shoots you down and says that she's not ready for dating yet, let her know that the offer still stands when she is.