During the trial, Aaron Campbell repeatedly denied he abducted, raped and killed the six-year-old on the Isle of Bute last July.
However, he admitted his offences to a psychologist preparing a report to the court ahead of his sentencing.
Judge Lord Matthews imposed a life sentence for the 'brutal' murder.
He said Alesha had been violated and murdered and told Campbell he was a 'cold, callous, calculating, remorseless and dangerous individual'.
The judge said the teenager had shown complete arrogance during the trial and had failed to plead guilty despite overwhelming evidence against him.
He said he had shown 'a staggering lack of remorse'.
"Not once did I detect a flicker of emotion from you," he said.
Alesha, who was staying with her dad and gran at their home in Rothesay, Isle of Bute, was reported as missing at around 6.25am on the morning of 2 July.
Her body was then discovered by police in the grounds of a derelict hotel around half a mile away at 9am.
According to reports from Metro, Campbell had told his social worker that 'it took everything to stop laughing' and he was 'quite satisfied with the murder.'
The court was told Campbell was 'entertained' by the thought of rape and had wanted to do 'something excessive' for at least 12 months before he killed the little girl. He had drank a bottle and a half of wine when he crept into Alesha's father's home in search of cannabis.
After taking her to local woodlands and committing his crimes, he went to a nearby beach, got rid of his bloody clothes, before returning to the crime scene to get his mobile phone.
A police officer who had arrived at the scene of the crime wept as he remembered what he saw.
During the trial, the court heard that the little girl had suffered 117 injuries, with a post-mortem finding she died from 'significant and forceful pressure to her neck and face'.
A forensic expert told the court that DNA matching Campbell's was found on Alesha's body and clothes, with some samples at odds of a billion-to-one of being his.
The jury at Glasgow High Court took only three hours to find him unanimously guilty following a nine-day trial last month, in which he persistently claimed he was innocent.
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