The memo shows that Tony Blair warned the Home Secretary of the dangers of "a sense of fatalism" creeping in over yob culture.
It says Mr Blair told Mr Clarke at a meeting on 20 June that it was vital the Government addressed low-level disorder and misbehaviour, which he said were seen as "a real issue" by the public. He ordered urgent action to prevent a "sense of fatalism" setting in.
Mr Clarke is seen by some as having a less authoritarian stance on crime and disorder issues than Mr Blair and his predecessor, David Blunkett. Mr Clarke is said to be keen on tackling crime and its causes, rather than introducing a raft of new punishments.
But the memo shows that Mr Blair has ordered Louise Casey, who takes a hardline stance, and heads the Government's anti-social behaviour unit, to report personally to him in future - as well as her immediate boss, Charles Clarke. "In addition Louise Casey should report regularly and directly to the Home Secretary and the Prime Minister.
The Home Secretary said that he would write to the PM setting out how this would work in practice the next few days," the memo says.
The Prime Minister also expressed frustration about the handling of ID cards and prisons. He said that the ID cards debate has been dominated by rows over the cost of the cards. He told Mr Clarke to announce a cap of around £100 by September.
The Prime Minister also said he wanted an end to the culture where decisions on whether offenders go to prison are led by the availability of prison places, rather than whether they deserved to be locked up.
Downing Street insisted yesterday that Mr Blair has "complete confidence " in his Home Secretary. A spokesman said any suggestion that the meeting indicated the PM was unhappy with Mr Clarke's handling of the " respect agenda" was "absurd".
"While we don't comment on leaked documents, it shouldn't come as a surprise to anybody that the Prime Minister and Home Secretary discuss such issues," he said.
Public support for ID cards, however, has plummeted. A YouGov poll in The Daily Telegraph today which found that backing for the scheme fell from 78 per cent less than two years ago to 45 per cent.
The survey suggests that voters are increasingly fearful about the costs of ID cards and the prospects of administrative chaos that could follow their introduction. Only one in 10 surveyed believes the cards would be introduced efficiently.
Last week a report by the London School of Economics said the cost of the cards could be as high as £20bn, three times the government's estimates.
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