I'd just wait until she's outside, possibly early evening, ask her if she's been to so and so to eat, if she says no, ask her if she'd like to go try it. If she has been to whatever place, ask her if she'd like to go back. I've found asking them to go eat takes away all awkwardness.
Followup, even better, wait to ask if you think you can hold a 15 minute conversation, just throw the question right there in the middle of it. You get your date and she'll be much more inclined to say yes if she knows she'll have to go to continue the conversation.
No I haven't made a move yet but I will be sure to! I'm not scared unlike what that dickweed SacTown says
Make plans to have a get together or party at your place, but only invite friends who are less attractive than you are. You don't need any competition. Several days in advance, go over to her place and introduce yourself (if you haven't already) and tell her when you'll be having the party and to let you know if you get too loud or anything. Then casually mention that she's more than welcome to stop by and bring some friends if she wants. Since you live in such close quarters, there's a good chance she shows up.
OP, let me tell you something. Time can never mend the careless whispers of a good friend. To the heart and mind, ignorance is kind. There's no comfort in the truth. Pain is that all you'll find.