It happens every day at my workplace.
We have a few people who do nothing but talk all day, every day. A couple of them report to me. They do not do any “real” work. They just talk, and talk, and talk…
I learned the hard way that, if you report these people, you are the problem, and you will be told to take training classes. 
(Sorry for the hijack, but I’m still licking my wounds over it…)
I guess there are some places like that. I’ve mostly worked in small offices with the Big Boss on the premises. The Big Boss can talk all day if he wants. You, peon, on the other hand…
I’m still blown away that I’m allowed to listen to YouTube online as I work. But my work better be done, and done right.
To be fair, her job involves meetings and presentations, which she does attend without fail and preparatory stuff, which she can do from home (and is also the reason she stays at work until very late).
I’m sure she gets her work done.
Just ask her out. 
See above advice. 
All you need is one minute:
Her: Bonjour, Espace. Ca va bien?
You: Look, here’s the thing. I really enjoy our chats. Would you like to come on a date with me?
(then suggest something simple like coffee or visiting a museum…)
My opinion is that she is using you, perhaps not consciously but using you all the same. She gets to flirt, chat, gets attention, support, all those nice little things that stroke the ego, but push come to shove she doesn’t actually want anything more than that. My guess is that if you actually asked her out she would dangle you along and there would probably be lots of deep and heartfelt conversations about conflicted emotions etc etc, but that it wouldn’t actually go anywhere.
Of course, a more confident man wouldn’t care about my opinion and you shouldn’t either. Ask her out or don’t, its up to you, just don’t let her live rent free in your head for so long like you are doing.
Anyone else got Bonnie Raitt’s “Something To Talk About” going through their heads? 
This. It’s time to find out if she’s available and interested, or not. If she is, time to take the romance offsite. If she’s not, I’d let her know that while you enjoy your professional relationship with her, she is coming across as interested, and while you aren’t telling her to not come around, you’d like her to dial it back a bit.
There’s always that risk, of course, but my office is filled with married couples who met here. The main thing is to keep any relationship drama off the premises.
ETA:
From the last paragraph of post #7, it sounds like this has all happened this month. If so, it hasn’t been that long. But yeah, time to have that conversation and either turn it into something more, or something less.
I object to that characterization. She’s presumably listening to him, flattering him, paying attention to him, too.
But yes, if you want more from this relationship, the time has come to clarify the situation.
I could swear I read almost this exact thread a few months ago, but now it’s gone.
Put me in the “just ask her out” bin. Having a serious talk abut “clarifying” the situation sounds like something from a bad rom-com. “Would you like to go out to dinner with me sometime?” will get you all the answers you need. If she declines, fine. No more really needs to be said.
I’ve worked with many people to whom I have been attracted. Some of them gave indications that they wanted a meaningful FRIENDSHIP, and nothing more. Others would state, quite frankly, that they would like to, but they had other another relationship at the time. Finally, I married a co-worker (who, BTW, had been trying to set me up with her best friend…funny that).
Maybe she just needs and wants a friend, and she has a naturally flirty manner around men and friendships. God knows I had that problem in my 20s. I would find a friend at work - male - someone I saw as a friend - maybe a mentor or father figure - he’d interpret my interest in him as romantic.
Assume she wants a friend. At some point you can refer to your romantic interest in her in an indirect way “I’d ask you out if it weren’t for mystery man and us being coworkers …” and see how she reacts. She might react with “mystery man is over, and I don’t see why being coworkers should stop us” - or she might react with “oh…well, there is mystery man and we are coworkers” which you should interpret as “yeah, I’m not into you that way.” Or she might say the second and then ask you out in six months herself when she is in a different emotional space.
Just. Ask. Her. Out. (cf. Gordian Knot)
Thank God; I thought I was losing my mind. It came up in this thread last July:
https://boards.straightdope.com/sdmb/showpost.php?p=21733506&postcount=48
And therein references developments as far back as last November, with the implication being that it’s been going on longer than that.
Dude.
Just.
Ask.
Her.
On.
A.
Date.
Don’t call it a date. Just ask her if she wants go do a happy hour with snacks, or get some dinner, after work. That way, if she’s been stringing you along and shuts you down, you’ll save face.
Seriously?? The same guy?? From FOUR MONTHS ago? 
Actually, now that I reread the OP, it sounds like this is maybe a different woman? With whom OP is in the exact same situation?
Hmmm.
Is the OP writing a novel?
Yes, in the exact same situation.
You said it better than I could.
I had a rather long reply typed out, but I’d like to hear an explanation first.
Good to know that the pace has picked up considerable since then!
Wait a minute. As of this past August, OP had a wife?
https://boards.straightdope.com/sdmb/showpost.php?p=21803215&postcount=1
They got married quickly! Then divorced! Then the next one came along!