It was a euphoric day, and rightly so. But for all the celebrations, it does not change the hard fact that many political obstacles still lie ahead.
Chief among them is the task of creating a transitional Iraqi government which will allow President George Bush formally to declare an end to the US occupation while keeping some US troops in Iraq, their presence ensured by an agreement with the new US-sponsored administration.
The Americans and their allies have set a target date of 1 July for a handover of sovereignty to an Iraqi transitional authority, paving the way for a new constitution and, eventually, general elections.
That seemed an ambitious goal when the US and the US-appointed Iraqi Governing Council unveiled these plans in mid-November. The exact means of appointing members of the transitional authority is still the subject of debate. And, one month on, the timetable still seems ambitious.
Much depends on the tactics of the Iraqi insurgency and the anti-American sentiment which underpins it.
The capture of Saddam Hussein will not bring an end to the resentment felt by many Iraqis - among them plenty of Saddam-haters - about the mass unemployment that followed the US-led invasion; about the deluge of western consumer goods in Baghdad that they cannot afford; about the humiliation and deadly violence visited upon them by ill-trained and trigger-happy American troops; about the useless public services, or the risk to women who venture in the evenings on to the streets, or the kilometre-long queues to buy petrol.
All these issues are still in the equation, generating anger and instability and complicating efforts towards reconstruction and political progress.
With Saddam's capture, the Bush administration has scored an undoubted political triumph, one which is likely to tempt it to explore the idea of speeding up the transfer of sovereignty so that it can declare a victory well before the US presidential elections. It hopes that such a transfer of power will reduce the bloodshed, which means that it can get some of its soldiers out of the firing line.
But the US must still settle some outstanding issues before it can expect to create a plausible interim government which satisfies the aspirations of a divided country and offers the 25 million Iraqi population the prospect of a peaceful future.
To do this it must establish new Iraqi security forces which are sufficiently large and competent to defend the new governing authority, albeit it with continued support from American and Allied forces. These Iraqi forces must be capable of matching relatively well-trained guerrilla cells, including suicide bombers.
And - if matters ever reach this stage - they must be able to provide enough stability for a general election to take place.
The arrest of Saddam does not change the fact that the Americans and the Iraqi Governing Council must devise a means of creating a transitional government which satisfies demands from within the Shia majority for a body that is as democratic as possible - a point consistently pressed from behind the scenes by the Shia community's most influential religious leader, the Grand Ayatollah Ali al-Sistani.
They must also find a way to allay Sunni fears of being excluded from power - a concern that is widely seen as one of the factors driving the insurgent attacks, most of which has been concentrated in the Sunni Triangle.
The task of building new Iraqi security services, which are vital to the long-term success of a new administration, has so far been blighted by repeated suicide bombings.
The new Iraqi army has also got off to a dismal start, with the resignation of 300 of the first intake of 700 soldiers. The US military has attributed the latter development to unhappiness over pay, although fear of being recognised and killed as an American collaborator is clearly a major factor. If the resistance tails away, then recruitment to the ranks may be easier.
But in this, as in much else, there is still a mountain to climb, which remains no less formidable than it was before Saddam was dug out of his wretched little hole.