With their troops at the airport and two powerful thrusts about to arrive at the city outskirts, Baghdad lies at the mercy of the Americans.
Attention now turns to the issue of what they will do next.
The problems of pacification are already at hand. And they are not just military. This is a political and diplomatic war as well.
How the end is handled will help determine Iraq's future and the future of relations between the Arab world and the United States and Britain.
The US troops could surround and squeeze the capital, just as the British are doing to Basra. That would take time.
They could seize the moment and drive for the heart of the city, hoping to precipitate the end. But they risk getting attacked as they were in other cities and there is the problem of civilian casualties which they want to avoid.
A sudden entry is unlikely unless the Iraqi military collapses.
Nobody knows what kind of resistance is likely in the streets of Baghdad. The Republican Guard did little on the roads outside. But some fighting has to be assumed.
There must also be a lingering hope that somehow the government of Saddam Hussein will implode. The US might try to provoke such a collapse, through a coup perhaps.
In Washington the Chairman of the Joint Chiefs of Staff General Richard Myers floated the idea of taking over certain parts of the city after probing attacks and patrols, especially with special forces, and setting up an "interim administration."
The US Defence Secretary Donald Rumsfeld is reported to have suggested setting up such an administration in the south of Iraq at least.
The Washington Post quoted US officials as saying that they might just declare victory even if Saddam Hussein has not been overthrown.
The plan would be to "recognize a moment when the military and political balance tilt decisively away from Hussein's Baath Party government."
The plan sounds like a variation on the old prescription for Vietnam - "Declare victory and leave". This is "Declare victory and stay".
If such a policy were followed, the benefit might be that it encouraged elements of the population to rally against the regime.
Start with Shia areas
US officials are also interested in developing links with the Shia population as a way of undermining Saddam Hussein who has repressed them.
The Shia populated parts of Baghdad could be the starting point for patrols to see if the people are receptive. Iraqi exiles however caution that the people in Baghdad are often mixed up and cannot easily be separated. And even when they can, it doesn't follow that they take a particular view.
US forces fear a chemical attack
The experience of the US 101st Airborne Division in the Shia holy city of Najaf could be instructive. The Shia need careful handling but are not necessarily hostile.
Troops ran into an angry crowd when they approached the revered Imam Ali shrine but local religious leaders calmed the crowd and the soldiers themselves backed off.
US commanders claim that the Shia leadership is now telling the people not to obstruct the soldiers.
If true, that would significantly help the process of pacification.