I pass by this girl once a week, but she's always busy. We aren't friends or anything. Would it be weird to ask her out?

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If you can take better advantage of the regular contact you’re already getting, you’re much better off doing that before asking for a date, instead of asking for a date in order to possibly get more regular contact.

Only ask for a date when you’re reasonably sure you’ll get a yes. Your task is to get to know her well enough during your once-a-week meetings and make a good enough impression that she is going to want to spend more time with you, not just the other way around.

If I were in your shoes, I’d simply wave hi every time I see her. I might find something light to say just to break the ice and see if I can entice her into a short conversation about nothing serious. The next time I see her I’d get a little closer by asking how her week went, all depending on just how interested she seemed.

If she opens up I’ll reciprocate, and seeing as we’ve gotten to know each other a little bit, I might suggest, as a teaser just to see how she might feel about me, some small, short excursion if she has a few minutes. Like if I saw something weird I’d offer to take her to go see it. Just something to break her out of her routine a bit that focuses on us spending time together.

The week after that, that’s when I might ask her on a easy, breezy date like grabbing coffee.