By Sarah Toms
BBC News, Washington
UK Prime Minister Tony Blair announced his timetable for pulling troops out of Iraq as thousands more US soldiers were arriving to restore order in Baghdad.
Mr Cheney said the UK withdrawal showed how well things were going
The UK has long been the most prominent US ally in Iraq, but the US expressed no dismay at the British withdrawal.
Vice-President Dick Cheney put a positive spin on Mr Blair's action, saying it reflects success in bringing stability to some areas of Iraq.
It was "an affirmation that in parts of Iraq things are going pretty well."
"I talked to a friend the other day who had driven from Baghdad to Basra in seven hours and found the situation dramatically improved compared to where it was a year or so ago," he told ABC News from Japan.
Mr Blair announced on Wednesday the UK would withdraw 1,600 British troops in the next few months from Basra.
US Secretary of State Condoleezza Rice said the coalition remained intact, pointing out that thousands of British troops would stay in southern Iraq.
"It is the plan that as it is possible to transfer responsibility to the Iraqis, that coalition forces would no longer be needed in those circumstances," she said in Germany.
While British troops have seen their fair share of combat, security analysts say the mainly Shia south has been easier to police than other areas with more sectarian tension.
The White House said the pullout of British troops was something America could copy once sectarian violence had been dealt with in Baghdad.
But with the British announcement coming as the Americans send in extra troops to Iraq, some analysts see signs of a strategic split between the two allies.
Since the US-led invasion in 2003, more than 3,000 US troops and tens of thousands of Iraqis have been killed. More than 130 British troops have also been killed.
The tide of American opinion has turned against the war and President George W Bush is facing strong opposition in Congress, which is now controlled by his opponents from the Democratic party.
Democrats in the US House of Representatives approved a non-binding resolution last week opposing the troop increase.
The British decision to pull out troops will only add to the pressure on President Bush to bring home American soldiers as well.
But Mr Cheney has remained firm, saying Washington would not back "a policy of retreat".
"We want to complete the mission, we want to get it done right and we want to return with honour," he said.