By Kevin Connolly
BBC News, Washington
Rarely can a military campaign have ended so ambiguously and with so little fanfare.
On the campaign trail Mr Obama pledged to remove troops in 16 months
America's combat mission in Iraq, which cost thousands of lives and billions of dollars, which broke homes and hearts, which angered enemies and alienated allies, will end by August 2010.
That means that tens of thousands of soldiers serving in what the US military calls "combat brigades" will be home 19 months from now.
America and Iraq have a bilateral agreement which envisages the withdrawal of all US forces by the end of 2011.
But as befits a conflict whose origins remain a matter of dispute and whose goals have changed dramatically since it was first embarked upon, there is a degree of uncertainty in all this.
America will retain a substantial residual force in Iraq for many months after the August 2010 deadline, and the scope of its mission is far from clear.
The original goal, of course, was to find the weapons of mass destruction (WMD) which George W Bush and Tony Blair insisted were somewhere in Iraq under the control of Saddam Hussein.
When the WMD proved to be a figment of someone's imagination, both the US and UK tended to point to the removal of the Iraqi leader as a sort of retrospective justification in itself, even though they could not have used regime change as a justification before the conflict began.
Briefly, the US dreamed of constructing a new Iraq which would serve as a kind of beacon of democracy in the Middle East - a different kind of nation in the region which would act as an inspiration for wider change.
Now it is clear that Washington (and its allies) would be more than happy to settle for the goal of leaving behind a relatively stable place which does not threaten either its neighbours, or US interests.
It is a modest goal when you think of all that money and all those lives - but at least it is probably attainable.
In US domestic terms, Barack Obama has delivered on a campaign promise to "end the war" in Iraq - but he is probably ending it in a way that very few of his supporters who heard and cheered that pledge would have anticipated.
At the moment, for example, the US has 142,000 soldiers in Iraq - Mr Obama spoke during the campaign of removing them within 16 months of taking office.
Once elected, he asked his military advisers to prepare various different scenarios for "drawing down" the force and they offered a choice of timetables - 16 months, 19 months or 23 months.
Mr Obama, who is moving much more cautiously on diplomatic issues than he is moving on the economy, chose the middle option.
The residual force that remains after that first deadline next year will have what is called a "training and support" role - a vague catch-all term that can cover any military activity from teaching people to drive trucks all the way through to flying close air support for combat operations.
The phraseology suggests a force made up almost entirely of cooks, clerks and instructors, but a military establishment of up to 50,000 in a still uncertain security environment is certain to be left packing quite a bit of firepower.
Note of caution
Some Democrats are already irritated by the scale and length of that residual deployment, which was not mentioned in Mr Obama's campaign speeches, but he is adamant that he is doing the right thing.
"The United States will pursue a new strategy to end the war in Iraq through a transition to full Iraqi responsibility," he said in his speech outlining the plan.
Some 50,000 residual troops would remain in Iraq after 2010
America's goal, he added, was "an Iraq that is sovereign, stable, and self-reliant", and to achieve that aim, he insisted, the US would "work to promote an Iraqi government that is just, representative, and accountable, and that provides neither support nor safe-haven to terrorists".
But he also sounded a note of caution, in a passage underlining the very real threats faced by the country.
"We cannot rid Iraq of all who oppose America or sympathise with our adversaries," he said.
"We cannot police Iraq's streets until they are completely safe, nor stay until Iraq's union is perfected."
In a section of the speech aimed directly at the Iraqi people, he called on ordinary Iraqis to reject "those... who will insist that Iraq's differences cannot be reconciled without more killing".
And he reminded Iraqis that the US, too, had experienced "forces that destroy nations and lead only to despair".
"We endured the pain of Civil War, and bitter divisions of region and race. But hostility and hatred are no match for justice; they offer no pathway to peace; and they must not stand between the people of Iraq and a future of reconciliation and hope."
Feeling his way?
So, Iraq will soon be left to face the dangers and uncertainties of the future on its own - surely it is hard to imagine any government which emerges in the coming two years negotiating to allow that US "residual force" to establish a more permanent presence.
Before the argument about how and why we got into that war is over, we will be debating the long-term effects of the US presence and evaluating the safety, stability and decency of the country it leaves behind.
And one final thought. On the domestic economy, Mr Obama has made a dramatic start with bold colossal gestures like the bailout package, the stimulus programme and the budget.
On Iraq, he chose the middle of the three options for withdrawal - almost as though he is feeling his way on the international stage.
After the turbulence of the Bush years, it's a caution many Americans may welcome.