Kent County Council said the continuing flow of unaccompanied asylum-seeking children (UASC) from across the English Channel had left it with no choice.
The authority currently has 924 such children in its care, compared with nearly 630 at the start of last August.
Councillor Peter Oakford said it was "not a position we want to be in".
The cabinet member for specialist children's services said Kent County Council had seen a 30% rise in looked-after children in the past seven months.
Other authorities elsewhere in the UK have accepted full responsibility for 56 unaccompanied asylum-seeking children.
Mr Oakford told the children's social care and health cabinet committee: "This has affected our ability to place citizen children within Kent ourselves.
"We have had to place Kent children outside of Kent due to the influx of UASC, which is not a good position to be in and is not a position we want to be in.
"It's actually costing us more financially because we have had to place, I think, six children into residential care either short-term or longer term, which is far more expensive than normal foster care."
The council's services have been stretched since the start of the migrant crisis last summer as Kent is the closest British county to Calais and Dunkirk.
The number of child asylum seekers coming into its care has slowed to about 15 a week during the winter, but Mr Oakford said if that increased with the arrival of the warmer months the authority would be "in significant difficulties".

The Grocer magazine reported that the firm is charging £2.64 for a 250g jar.
It is the first supermarket to raise the price of Marmite since the recent dispute between Tesco and manufacturer Unilever over the effect of the falling pound on production costs.
A Morrisons spokesman said: "Sometimes we have to increase prices as a result of costs rising."
Unilever has raised the wholesale price of many household products, including PG Tips tea and Pot Noodles, because falls in the value of sterling have increased the cost of products made outside the UK.
The pound has lost nearly a fifth of its value since the UK voted to leave the EU in a referendum in June. It is the world's worst-performing currency on the global money markets this year.
This makes imported goods more expensive because they continue to cost the same in dollars or euros, but the price is higher when converted into sterling.
Morrisons said it did its best to avoid putting prices up. It added: "More often than not we have been reducing prices and more than 3,000 products are currently cheaper in our supermarkets than they were last year."
Tesco's row with Unilever broke out earlier this month and has since been resolved, according to both sides.
However, the Grocer reported that both Tesco and Asda have raised prices on a number of Unilever products, and Morrisons has also increased the prices of other items made by the company.
According to supermarket websites, a 250g pot of Marmite costs £2.64 at Morrisons, £2.50 at Sainsbury's, £2.35 at both Waitrose and Tesco and is on special at Asda for £2.
Vulnerable children are being placed in care outside their home county of Kent due to the influx of child asylum seekers, according to council chiefs.
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Supermarket chain Morrisons has increased the cost of a jar of Marmite by 12.5%, say retail industry experts.