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Mary Bell is said to have strangled the boys, aged four and three, "solely for the pleasure and excitement of killing".
The jury heard Mary, also known as May, was suffering from diminished responsibility at the time and therefore found her not guilty of murder.
Her accomplice, known only as Norma, aged 13, who had been jointly charged with Mary, was acquitted.
Very grave risk
As the verdict was read out, Mary broke down and wept.
Mr Justice Cusack described her as dangerous and said there was a "very grave risk to other children if she is not closely watched".
Mary's mother and grandmother, who were sitting behind her on the benches, also wept when the verdict was announced.
Martin Brown, aged four, of Scotswood in Newcastle was found dead in a derelict house on 25 May. The body of Brian Howe, three, also of Scotswood, was found on waste ground near his home two months later.
The two girls, who were playmates, also lived in the Scotswood area of Newcastle. They denied the charges.
The court had earlier heard Norma give evidence in which she described how Mary had tried to strangle Brian Howe. She said Mary ignored her pleas to stop hurting the boy so she left them and next time she saw Mary she was on her own with Brian's dog.
Jurors were told despite the age difference, Mary was the more dominant personality with a very worldly attitude.
Rudolph Lyons QC said: "For example, when she was being questioned by a detective chief inspector about a charge of murder she said to him, 'I'll phone for some solicitors, they will get me out. This is being brainwashed."
He said she also tried to throw suspicion onto an innocent boy in a "very cunning and insidious manner".
He continued: "Both girls well knew that what they did was wrong and what the results would be."
Home Office psychiatrist Dr David Westbury told the court Mary had a psychopathic disorder for which she needed treatment.
The judge said: "It is a most unhappy thing that, in all the resources of this country, it appears that there is no hospital available that is suitable for the accommodation of this girl."
Mary is being held at a remand centre. It seems likely she will be sent to an approved school where she will be held in a secure unit.
Mary was sent to Red Bank approved school in Newton-le-Willows in Lancashire.
The special security unit of the school, which accommodates more than 200 boys, was adapted to take her in February 1969.
Although there were concerns she was being starved of female company, it was felt too disruptive to move her to a new centre for "severely maladjusted children" which opened at Brentwood in Essex in 1972.
Her MP at the time, Labour's Robert Brown, said he was satisified she was receiving psychiatric supervision and full-time education. He said: "She is turning into a very presentable young woman and a very bright young woman indeed."
Mary Bell was released on licence in 1980. She was given a new identity and has since had a daughter.
Her case raised controversy again when her biography was published in 1998 and it emerged she had been paid for contributing to the book.
When her daughter reached 18 in May 2002 their right to anonymity could have been lifted.
However, the High Court granted them both lifelong anonymity under the Human Rights Act, in May 2003, on the grounds they were entitled to a private and family life.
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