The news was confirmed to the BBC by the Iraqi deputy foreign minister.
Iraqi TV said the execution took place just before 0600 local time (0300GMT). A representative of the prime minister and a Sunni Muslim cleric were present.
It is unclear whether Saddam Hussein’s half-brother and a former Iraqi chief judge were also executed.
All three were sentenced to death by an Iraqi court on 5 November after a year-long trial over the 1982 killings of 148 Shias in the town of Dujail.
A small group of Iraqis including a representative of the prime minister and a Sunni Muslim cleric were brought to a building somewhere outside the Green Zone and watched as the sentence was read out to Saddam Hussein, the BBC’s John Simpson says.
The former Iraqi leader was carrying a copy of the Koran and asked for it to be given to a friend.
The noose was then placed around his neck and his face was hooded before the trap door was released. The execution took just a few minutes.
Video footage of the execution is expected to be released as final proof of Saddam Hussein’s demise although it is expected to stop short of showing the actual death, our correspondent says.
US troops and Iraqi security forces are on high alert for any violent backlash. The US State Department has urged all its embassies to increase security.
‘End of a dark period’
News of Saddam Hussein’s execution was broadcast on state-run Iraqiya television, as patriotic music and images of national monuments were played out.
|It is an important milestone on Iraq’s course to becoming a democracy that can govern, sustain, and defend itself
US President George W Bush
Saddam Hussein was hanged first, followed by Barzan and then Bandar, it announced. However, an Iraqi national security advisor has since said only Saddam Hussein was hanged.
A scrolling headline read: "Saddam’s execution marks the end of a dark period of Iraq’s history."
Other Arab TV stations aired live footage of the sunrise over Baghdad’s Firdous Square, where US Marines pulled down a statue of Saddam Hussein, after he was deposed in April 2003.
The BBC’s Peter Greste in Baghdad says Shias have generally welcomed Saddam Hussein’s death and hailed the execution as justice for the suffering endured under his leadership.
But Saddam’s own Sunni tribesman were angered by his treatment and may well protest once more, our correspondent adds.
‘Held to account’
US President George W Bush hailed the execution as "an important milestone" on the road to building an Iraqi democracy, but warned it would not end the deadly violence there.
"It is an important milestone on Iraq’s course to becoming a democracy that can govern, sustain, and defend itself, and be an ally in the War on Terror."
UK Foreign Secretary Margaret Beckett welcomed the fact that Saddam Hussein had been tried by an Iraqi court "for at least some of the appalling crimes he committed" and said "he has now been held to account".
France called on Iraqis to "look towards the future and work towards reconciliation and national unity".
Published: 2006/12/30 05:43:05 GMT