The government and AIG restructured the terms of the giant insurer’s bailout Thursday for the umpteenth time. The deal could let Federal Reserve chief Ben Bernanke more or less wash his hands of the AIG mess within six months — which is probably just as well, since he has a few other things on his plate.

But Treasury Secretary Tim Geithner — who was running the Federal Reserve Bank of New York two years ago when it plunged headfirst into this morass — will have his work cut out for him spinning an ever-increasing pile of AIG straw into taxpayer-reimbursing gold.

AIG chief Robert Benmosche called Thursday’s pact, which will convert Treasury’s $49 billion in TARP preferred shares to common stock that can be sold into the market, a “pivotal milestone as we deliver on our long-standing promise to repay taxpayers.”

Geithner says the agreement “dramatically accelerates the timeline for AIG’s repayment and puts taxpayers in a considerably stronger position to recoup our investment in the company.” But it’s early yet for taxpayers to declare victory, given persistent questions about AIG’s health and official estimates made this year that the bailout will end up costing taxpayers $30 billion or more.

The real winners in this deal are AIG investors, whose stake in the company will get diluted less than anyone might have expected, and the Fed, which will have to be made whole on most of its AIG-related loans before the Treasury can start to cash in on its own so-called investment.

Right now, AIG has $20 billion in borrowings from the Federal Reserve Bank of New York. The New York Fed also owns $26 billion in preferred stock in two AIG insurance companies.

Under the deal outlined Thursday, AIG will repay the Fed credit line with resources on hand as well as the proceeds of various planned asset sales. AIG said it expects to repay the $20 billion — which it must do before it can convert the Treasury preferred shares to common — by the end of the first quarter.

The company will then actually expand its Treasury borrowings by $22 billion to buy out most of the Fed holdings in the Alico and Aurora units. AIG will apply the proceeds of ongoing asset sales to buy out the rest of the Fed’s holdings in these so-called special purpose vehicles, or SPVs.

Whenever these transactions close, the Fed’s direct exposure to AIG — which started with an $85 billion loan the evening of Sept. 16, 2008, then climbed to $123 billion with a plan the next month to unwind the company’s disastrous securities lending practice — will be down to zero.

The Fed will still hold some $30 billion in loans made to unwind two of AIG’s most troubled portfolios — its disastrous foray into securities lending and the payoff of the big banks that held collateralized debt obligations guaranteed by AIG. The Fed has been showing gains on those loans, though we are years from a final reckoning.

The costs of these decisions aren’t just financial, of course. The CDO buyout is perhaps the most toxic government program of the financial meltdown, having provoked accusations that the Fed was covering up a backdoor bailout of the big financial institutions both here and overseas that traded with AIG. TARP is widely abused in spite of its apparent financial success.

And while Thursday’s deal means the Fed soon will be mostly done with AIG, Treasury will find itself in even deeper.

It will hold a 92% stake in the company, up from just under 80% now — in addition to $26 billion worth of preferred stock in the AIG insurance companies.

Turning all this paper into cold, hard cash won’t be simple. The government has been able to whittle down much of its Citigroup stake this year without upsetting the market or undermining the value of its remaining shares in the company.

But the government never held such a large stake in Citi, and even so Treasury’s yearlong stock selloff is running behind schedule.

So Geithner has his hands full with this one. Somehow, that has a familiar ring to it.