Howard Hughes had to be restrained by four security guards after losing his appeal against conviction for "the crime that broke the nation's heart" last Thursday.
The Lord Chief Justice, Lord Bingham, said the case against Hughes was "strengthened immeasurably" after he told his dad where the murdered school girl's nightdress was.
"The perpetrator would be best placed to know where the clothes of the child were," said Lord Bingham, at the Court of Appeal in London on Thursday.
Hughes, 32, was found guilty and received three life sentences at Chester Crown Court in July 1996.
At the time, the judge, Mr Justice Curtis, described him as a "fiend" who should never be freed.
But Hughes, known in his native North Wales as Mad Howard, continued to protest his innocence - and begun a letter-writing campaign from his cell shortly before his appeal, which began last Wednesday.
Representing Hughes, Mr Patrick Harrington QC, said it would be "difficult" to persuade the court the conviction was unsafe.
But he said there were some "deeply troublesome aspects" about the case.
Seven-year-old Sophie was abducted from a tent in the back garden of her uncle's home in Llandudno during the summer of 1995.
Hughes, who lived in Colwyn Bay, was seen in Llandudno over the weekend of Sophie's disappearance and the discovery of her body on the beach.
"The police took no time to decide who they wanted," said Mr Harrington. "Hughes was well known in the area. He led a strange and bizarre life."
But, said Mr Harrington, there was no evidence on Sophie's body to connect Hughes with the crime.
And Hughes' counsel also questioned the evidence of two of the prosecution's witnesses.
But his main concern was that police failed to tape a conversation between Hughes and his estranged father Gerald.
Hughes' father told police his son confessed to the crime during the meeting - and later told them where they could find Sophie's nightdress.
But Mr Gerald Elias QC, prosecuting, told Lord Bingham, sitting with Mr Justice Smedley and Mr Justice Thomas, the room wasn't bugged because the equipment had to come from Manchester.
He said the police had investigated the matter properly, finding "small pieces of a jigsaw that needed to be added together".
In reply, Mr Harrington said "the jigsaw puzzle was incomplete". He accused the police of "a rush to charge" Hughes because he had a mental handicap and a memory problem.
"The police knew he was a person who was vulnerable," he said. "He was never given the opportunity of challenging or denying what was said when he met his father."
Lord Bingham dismissed Hughes' appeal on Wednesday. When he explained why the following day, Hughes denied ever making a confession.
"Sir, I never made a confession. Sir, I never made a confession," he shouted from the dock.
Security guards tried to bundle him out of court but he stood his ground before smiling, nodding and saying: "Okay, okay."
The three appeal judges said they could not fault the handling of the trial, Hughes' detention and questioning or the identification of suspects.
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