Iraqi officials have shown journalists video footage of the hanging of two of Saddam Hussein's aides, during which one of the men was decapitated.
Iraqi officials said Barzan's beheading was an accident
The film shows Barzan Ibrahim - Saddam Hussein's half-brother - and Awad Hamed al-Bandar hanged side-by-side.
Barzan, former intelligence chief, and al-Bandar, former head of Iraq's Revolutionary Court, were convicted over the killing of 148 Shias in 1982.
The government said Barzan's decapitation was accidental.
The latest hangings drew expressions of concern from among the international community.
Speaking on a visit to Egypt, US Secretary of State Condoleezza Rice said while the executions were an Iraqi process, "we were disappointed there was not greater dignity given to the accused under these circumstances".
The UK prime minister's spokesman said it was "clearly wrong" if the executions had not been carried out in a dignified way.
Others called for the Iraqi authorities to end further executions and focus on national reconciliation.
Some of the strongest reaction came from the office of UN Secretary General Ban Ki-moon.
UN spokesman Michele Montas said Mr Ban "regrets that despite pleas from himself and the high commissioner for human rights to spare the lives of the two co-defendants, they were both executed".
The BBC's Andrew North in Baghdad says the video first shows both men being prepared for execution standing next to each other.
They were both dressed in orange boiler suits.
Executioners in balaclavas placed hoods round both men's heads, then the noose.
A short while later the footage, which is silent, shows both men fall.
Almost immediately the rope that was round Barzan's neck flicks upwards, the body dropping below.
The cameraman then shows the pit below and a headless body, bloodied at the neck and what officials say was Barzan's head still covered by a hood.
Al-Bandar's body was still hanging above, said one official who was present at the execution.
Our correspondent says officials say they are not planning to release the footage publicly.
The two men were later buried near Saddam Hussein's grave in the former Iraqi leader's home village of Awja about 180 km (110 miles) north of Baghdad.
Some 3,000 people, many firing guns and shouting slogans against the government, gathered at the site for the funerals, AFP news agency reported.
Witnesses said Barzan and al-Bandar were shaking with fear as they approached the gallows.
One of those present, public prosecutor Jaafar al-Moussawi, told the BBC that when the trap door opened, he could only see Barzan's rope dangling.
"I thought the convict Barzan Ibrahim al-Tikriti had escaped the noose. I shouted that he's escaped the noose, go down and look for him. I went down a few steps ahead of the others to see: I found out that his head had separated from his body."
The hangings took place at 0300 (0000 GMT), apparently in the same building in north Baghdad where Saddam Hussein was put to death on 30 December.
The manner of Saddam Hussein's execution drew international criticism after unofficial mobile phone footage showing him being taunted and insulted in his final moments was released.
Government spokesman Ali al-Dabbagh said there were no such scenes at the hanging of his aides.
The BBC's world affairs correspondent Mike Wooldridge says reaction in Baghdad to Barzan and al-Bandar's executions has been mixed.
He says residents of Baghdad's largest Shia district, Sadr City, have celebrated the latest hangings, especially Barzan's.
But other Baghdad residents have said the executions have nothing to do with the problems Iraqis face every day, our correspondent adds.
In the Shia holy city of Najaf, residents beat drums and marched in the streets at news of the executions.