Any new unaccompanied migrant children arriving on small boats will now be left with Border Force after Kent County Council announced it has reached capacity.
Large numbers of migrant children reaching the UK have led to an "unthinkable" situation where the council says it "cannot safely accommodate any more new arrivals".
More unaccompanied migrant children arriving over the weekend "tipped the balance", council leader Roger Gough said on Monday.
This is despite efforts to work with the Home Office and other local authorities, he said.
Now any new migrant children arriving unaccompanied in Kent will remain in the care of Border Force in Dover, the PA news agency has learned.
They will remain there until they can be transferred into the care of another UK local authority with capacity to safely care for them.
The Home Office said unaccompanied migrant children would be cared for in the Kent Intake Unit at Dover, which it described as a "processing centre".
Charities say the situation is "deeply worrying" and "entirely preventable", criticising Home Secretary Priti Patel's actions in recent weeks.
A spokesman for Kent County Council (KCC) told PA that the increased arrivals of unaccompanied migrant children has "exhausted" Kent's resources such as social workers, independent reviewing officers, care workers and suitable accommodation.
Meetings between the council and Government are under way in an effort to find a resolution today, the council said.
Bella Sankey, director of Detention Action, described the situation as a "scandal" and said it "should be a source of deep shame for this Government".
She said: "This is a political failure, pure and simple... The Government must urgently U-turn and ensure that vulnerable children are immediately taken into care rather than detained in prison-like facilities.
"It is deeply worrying that this entirely preventable situation has occurred.
"Instead of ensuring that unaccompanied child refugees are provided with essential support when they arrive in the UK, the Home Secretary has spent the last few weeks trashing the UK's proud record of helping the world's most vulnerable and trying to turn this situation into a Trumpian culture war.
"Priti Patel needs to show some moral leadership and quit the playground politics."
Migrant charity worker Bridget Chapman, of the Kent Refugee Action Network, said: "KCC have been warning for some time that without additional support from central government they would reach this point.
"Our main priority is to ensure that vulnerable children are properly cared for and we urge the Government to urgently work with KCC to find a way forward.
"We are now in a position whereby it's not clear what will happen to unaccompanied children arriving in Dover from now on. That is clearly unacceptable and needs to be resolved immediately."
A Home Office spokesperson said: "This is an unprecedented situation and we continue to work closely with the Department for Education and local government on provision for unaccompanied minors.
"Unaccompanied children arriving in Dover are being cared for in the Kent Intake Unit before being placed in appropriate social services care."
The Home Office said the Kent Intake Unit is a "processing centre" where people who arrive by small boat are taken, and rejected descriptions of it as a detention location.
It said discussions are continuing "at pace" with social services from other local authorities already on board.
The Home Office spokesperson added that it does not foresee any children remaining in Border Force care for very long.