Start with your manager. You're going to need a lot of support here, both in terms of budget and acceptance of risk. You'll need a budget to hire Barney's replacement as soon as you can. You'll need acceptance of risk because when you pull the plug and Barney is gone, some duties will fall through and be forgotten or mishandled -- probably fewer than you expect, but there will be some.
You'll hire Barney's replacement, officially as a staff analyst reporting directly to you. One qualification is discretion; your new hire should be clear that his/her future is to replace Barney and to keep this knowledge under his or her hat.
Once on board, your analyst's job (call her Betty) is simple: Figure out everything Barney produces -- simple to describe at least. She gets a copy of everything Barney produces and figures out how he produces it. She's free to bug Barney as much as she needs to. If Barney rolls his eyes, you let him know it's in his best interests; in the past he's expressed irritation when you've asked him to perform various types of analysis, so Betty will be taking it over. He just has to help her learn how to do it.
Give Betty three months (to allow her enough time and to make sure she was a good hire). Then reorganize, putting her in charge of Barney's organization and terminating Barney. He gets whatever termination package is customary at your company, but is escorted off the premises immediately.
It's possible Barney will leave in the middle of this process. Unlikely -- it's doubtful he'll find another opportunity that quickly and even more doubtful he'll depart with no other job in the offing.
But it could happen. If it does, let everyone know you're counting on them to pick up the pieces and figure things out as best they can. It's surprising how many responsibilities an employee like Barney can have that turn out, once they stop getting done, to be dispensible.