Prime Minister Theresa May has apologised to Caribbean leaders over the Windrush generation controversy, at a Downing Street meeting.
She said she was "genuinely sorry" about the anxiety caused by the Home Office threatening the children of Commonwealth citizens with deportation - writes bbc.com
The UK government "valued" the contribution they had made, she said, and they had a right to stay in the UK.
The PM said there was no "clampdown" on Commonwealth immigrants.
She said the current controversy had arisen because of new rules, introduced by her as home secretary, designed to indentify and expel illegal immigrants from the UK.
"This has resulted in some people, through no fault of their own, now needing to be able to evidence their immigration status," she told the foreign ministers and leaders of 12 Commonwealth nations in Downing Street.
The PM added: "Those who arrived from the Caribbean before 1973 and lived here permanently without significant periods of time away in the last 30 years have the right to remain in the UK.
"As do the vast majority of long-term residents who arrived later, and I don't want anybody to be in any doubt about their right to remain here in the United Kingdom."
Jamaica's Prime Minister Andrew Holness told Mrs May he looked forward to "a speedy implementation of your proposed solution".
He said it was "only fair" that people who had "significantly contributed" to the UK should be allowed to "take their place" as citizens.
The UK government has apologised after it emerged that some people who arrived from the Commonwealth decades ago as children were now being incorrectly identified as illegal immigrants.
The home secretary has announced a new taskforce to help those affected.
Mrs May had earlier watched as Mr Holness was cheered by the audience at the Commonwealth Heads of Government Meeting, as he told them: "As the case now stands and as history will show, citizens of the former colonies, particularly the West Indies, migrated to Great Britain, where they have significantly contributed to the building and enrichment of the country.
"Now these persons are not able to claim their place as citizens.
"Prime minister, we welcome your response and we look forward to a speedy implementation of your proposed solution.
"It is only fair and will lead to security certainly for those who have been affected. And it is the kind of inclusive prosperity for which we stand as Commonwealth peoples."
Cabinet Office minister David Lidington earlier said the government knew of no cases where British residents of the "Windrush generation" from the Caribbean had been deported because of their lack of documentation.
But he said officials were checking records to make sure nothing had "gone appallingly wrong in that way".
But Labour's David Lammy, who has led calls for action on the issue, tweeted that he had been made aware of a man who is due to be deported on Wednesday because he did not have the correct paperwork.
On Monday, Mr Lammy described the government's apology to the Windrush generation as a "day of national shame".
The Tottenham MP said it was "inhumane and cruel" that it had taken the government so long to act on the cases of those who arrived legally as children on their parents' passports but have never had their own passport.
Cases have emerged - such as these in the Guardian - of people who have lost their jobs, rights to health and benefits and faced deportation despite being able to show they have paid tax and National Insurance for decades.
On Monday, Home Secretary Amber Rudd accepted it was "appalling" that some of the Windrush generation faced deportation and promised they would be helped to attain the required documents for free. She added that she was concerned her department "sometimes loses sight" of individuals.
Thousands of people arrived in the UK as children in the first wave of Commonwealth immigration 70 years ago.
They are known as the Windrush generation - a reference to the ship, the Empire Windrush, that brought workers from the West Indies to Britain in 1948.
Under the 1971 Immigration Act, all Commonwealth citizens already living in the UK were given indefinite leave to remain.
However, the Home Office did not keep a record of those granted leave to remain or issue any paperwork confirming it, meaning it is difficult for the individuals to now prove they are in the UK legally.
Changes to immigration law in 2012, which requires people to have documentation to work, rent a property or access benefits, including healthcare, has highlighted the issue and left people fearful about their status.
Shadow home secretary Diane Abbott said the government should consider compensation for anyone who had been wrongly deported.
A Home Office spokeswoman said that any people who believed they or a family member had been wrongly deported should get in contact so their case could be reviewed.
The Migration Observatory at Oxford University estimates there are 500,000 people resident in the UK who were born in a Commonwealth country and arrived before 1971.
People born in Jamaica and other Caribbean countries are thought to be more affected than those from other Commonwealth nations, as they were more likely to arrive on their parents' passports without their own ID documents.
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