Kent County Council said it does not have the capacity for additional unaccompanied asylum-seeking children.
More than 400 children, most of whom have arrived in Dover after crossing the English Channel by small boat, have entered the authority's care so far this year.
The council said the responsibility should be shared across England.
Under-18s arriving in the county alone are passed into the care of the local authority, with a small number later transferred to other councils that volunteer to help.
County council leader Roger Gough said he warned the Home Office his authority "expected to reach safe capacity to meet its statutory duty of care this weekend".
The arrival of 13 more children in the past two days had "tipped the balance and the council simply cannot safely accommodate any more new arrivals," he added.
The Home Office has been contacted for comment.
It is unclear what will happen should more children arrive in the coming days.
"That is clearly unacceptable and needs to be resolved immediately," said Bridget Chapman, of Kent Refugee Action Network, which supports unaccompanied asylum-seeking children and refugees.
"Our main priority is to ensure that vulnerable children are properly cared for and we urge the government to urgently work with Kent County Council to find a way forward," she added.
In May, the government increased the amount given to councils to care for asylum-seeking children after Mr Gough warned social services in Kent were at risk of being overwhelmed.
Sue Chandler, cabinet member for children's services, said the voluntary national transfer scheme needed to be made mandatory.
While some children had been moved to other areas in recent months, "due to the continued high level of arrivals, it has not been enough to make a real difference to the numbers in Kent," she said.
Mr Gough has said the reduced amount of freight crossing the Channel due to coronavirus has lead to an increase in the number of asylum-seeking children arriving in Kent by boat.
Last week, 23 lone migrant children were taken into the council's care in a single day.
More than 4,800 people have crossed the English Channel in small boats this year.