There is an intern at my company who has been there for two months. During this time I've helped her briefly on a few projects and we have talked socially at work a little but mostly worked completely in different departments. She is of the same age as me and will not be staying in the company at the end of the internship. Would it be inappropriate for me to ask her if she would like to get a drink outside of work?
There shouldn't be any ethical conflict in asking her out, especially if she's about to leave the company. Ask her out first in a general rather than specific way (for example, "I would love to take you on a date sometime" as opposed to "do you want to have dinner on Thursday?") so that she clearly knows your intentions. That way, if she says no, you'll know it's because she's not interested, rather than because she isn't available on that day, and then there is no miscommunication/confusion about that.
Even if the intern is leaving the company soon, there might be a different power dynamic based on the different roles you had in the organization. So in addition to asking her out, it's very important to give her plenty of room to say no. For example: "I've really enjoyed working with you. This might sound forward, but I would love to see you again. Do you want to go out with me sometime? If not, I'd also be happy to hang out as friends." (Assuming that, for example, you are interested in pursuing a platonic friendship with her.)
The unethical part may be a thing when there is power or dependency in play (a boss asking out their reports for instance).
In larger companies this is something which is normed a way or another (a female friend of mine married a guy in her organization (he was her n-2) - once they started to date they were split into two organizations so that there is not formal hierarchical relationship and everything was good).
Some companies even encourage making a family with someone else in the company, or having several members of a family apply.
You are in a case where there is no hierarchical relationship between you (which is already easier) and on top of that she is leaving. So no - there is no problem here.
Now that I think of it I know a case where an ex-manager asked their intern out when that intern was leaving (to make sure all relationship has ended). Another happy marriage (at least it was a few years ago when I saw them for the last time).
UPDATE: Another case I recall is my professor at the university who married his student (he was a post-doc at that time and was teaching a group of students she was a member of). I got her as the TA for a course I was getting a few years later after they married.
It's risky asking her at work, particularly if she isn't interested—consider damage control. If you do ask her at work, make sure you don't have an audience.
Even then I advise against it, because when women are at work, they are hostage to the situation. FWIW this includes working women at their place of work, where you may just be a customer.
Get her a going-away card
I would do something different if I were in your shoes. I would get her a basic "Good Luck!" card. They sell them everywhere they sell greeting cards.. Inside the card, put something simple like:
"HER_NAME, you sure do stand out from the crowd!
Good luck on your next adventure
*Disclosure: I've never used note.ly, but I assume it allows people to share notes with others using short URLs similar to bit.ly, so to demonstrate this I formatted a fake note.ly URL and linked it to a bit.ly that's linked to note.ly's homepage. Hopefully you get the idea.
Make sure the note.ly note does something flirty like, suggesting you'd like to take her out, and 'Check one: yes, no, maybe.' The point is, make it fun. And don't forget to include your phone number in the note so she can actually respond!
Timing the card delivery is probably the hard part. Personally, on her last day, I'd take a half day off, deliver the card just before lunch, then bow out. This is mostly in the theme of "avoid an audience". Make sure it's a face-to-face hand-off, but don't dive deep; I'd say "HER_NAME, I've got to take off early today but I got you a going away card and didn't want to miss you. See you if I see you!" Then leave. Don't stall. Don't hesitate. Don't wait for additional conversation. Just go. As of now, your pieces are in play.
This solves some problems and does some fun things.
It doesn't put her on the spot at work, which saves everyone a whole mess if she isn't interested.
It's a little drama which will drive her crazy till she follows the link, "What?? What is this link?? What is he doing!?!?"
It's ultra memorable. And it might single-handedly reset whatever context/frame she has of you, which at bare minimum you need to catapult out of "coworker-zone" ASAP.
And best of all, it puts the ball in her court. If you hear from her at all after this, game on.
If she doesn't have a boyfriend and if she's at least 51% into you, I'm pretty sure this approach will at least get you a text. From there it's up to you. Hopefully I'm not too late. :) Good luck!