DNA confirmed that I (39m) have a daughter (18f), What now?

And yeah, I feel just as you would imagine. But it gets worse. She lives in the US, I'm in Europe. I don't know how to do this and really need advice.

So what happened is that months ago i got a call from a US number. Didn't answer since I don't know anyone there. Then again the same day, so I answered ready to yell at some scammer. A girl on the other side introduces herself and just say that she thinks I might be her father. That was the first thing she said. "Hi my name is Jane and I think you might my father." (fake name). Obviously I told her I don't have any children and that she has the wrong number. But then she drops a name I haven't heard in, you guessed it, 18 years! Back around that time I was in the US for a year on a job exchange with our sister company. Met a woman there and we hooked up over maybe a few months. I remember buying condoms after meeting her but I can't remember if we used them every time. Anyway I gave her my non-US number but never heard from her again.

The girl on the phone asks me if my name is OP and I freak and almost hang up on her. She tells me more about her mother and I just sit there like an idiot. She'd found my name and number in her mother's diary, and now as she turned 18 she wanted to contact me. She lives with her relatives now since her mother is unable to take care of herself for some reason. I still don't know the details there.

She ask me if I'd be willing to do an ancestry DNA test and even offered to pay for it. She ends by saying I should think about it and ask if she can call me again. I just say sure and we hang up. I swear I was about to faint right there and then. The whole conversation lasted only like 5 minutes.

I end up taking the test and we talk a few more times. The test took forever but yesterday she sent me a screenshot of her profile matching 50% to mine. I mean, that's just it right? We havent spoken since, just messaged some. I lied and told her I've been busy and we'll talk later, which now makes me feel like an ass. Up until this point I've kept it together, holding on to the thought that I'm probably not the dad, but now I can't deny it. Its probably been half my life since I last cried.

I haven't told anyone this. I was kinda hoping the DNA test would come back negative and then I wouldn't have to. How to even break this to my family? I just know they'll all want her number so they can call her and then they would start flying over to see her or bringing her here. They're really pushy like that. My mom is going to freak out. I don't see my family more than a few times a year, but we'll all get together during easter. Should I do it then with all of them at once or one at a time? I have a picture of Jane, but not much more info than what I've written here. I'm the youngest sibling and now I suddenly have the oldest of the cousins in the next generation. And I never wanted kids at all. I just don't like them and hate being tied up. Even had a vasectomy since then.

And that's the easy part. The real problem is what to say to Jane. Can some sane person give an example of what to say? I've promised to call her tomorrow.

Has anyone discovered they have an unknown child? If so, what did you do? How did it go? How did you connect? What happened when you met for the first time? What did your family say?

I'm a bit embarrassed by this, but is there any risk of me having to pay child support retroactively? I can probably afford it, but I don't really want to.

tldr: Got contacted by a young girl claiming to be my daughter, DNA later confirmed. Need help on how to proceed and what to say to her and my family.