As a result, Kate and Gerry McCann dropped their High Court bid to force disclosure of the information held by Leicestershire police.
The force has been involved in the search for Madeleine since shortly after she disappeared 14 months ago.
The parents, who wanted to avoid a legal conflict with the police or other agencies while at the same time obtaining as much material as possible to help in their continuing search, were content to withdraw their case.
Their spokesman Clarence Mitchell, speaking on the steps of the High Court in London, welcomed the move.
Mr and Mrs McCann were not present at the hearing in London. Mrs Justice Hogg said she had not asked for their attendance because "they have suffered enough and I wished to ease their burden".
The judge urged anyone with information about the missing girl to come forward.
"There is, of course, one person who knows what has happened to Madeleine and where she may be found," the judge said.
"I ponder about that person - whether that person has a heart and might understand what it must be like for Madeleine to have been taken and secreted from her parents and siblings and those she loved and felt secure with.
"I ponder whether that person has a conscience or any feeling of guilt or remorse or even cares about the hurt that has been caused to an innocent little girl, and whether that person has a faith or belief and what explanation or justification they might give to God.
"I entreat that person, whoever they may be, to show mercy and compassion and come forward now and tell us where Madeleine is to be found.
"I hope she will be found soon, alive and well."
The court heard that the McCanns, both 40, from Rothley, Leicestershire, were on holiday with their twins, Sean and Amelie.
Their spokesman, Clarence Mitchell, said outside court the couple would very much welcome the 80 pieces of information which the police had agreed to release.
"These are 80 potential new leads. We have information we can work on," he said.
The couple were also looking forward to much greater disclosure of information once their status as "arguidos" or formal suspects was, as expected, lifted by the Portuguese authorities.
Mr Mitchell said: "They will be strengthened and touched by what Mrs Justice Hogg said.
"It was immensely kind. In the context of the courtroom, it was a statement she did not have to make. It was extremely supportive and very generous and kind.
"I know Kate and Gerry will be incredibly touched."
Madeleine vanished from the family's holiday apartment in the Portuguese resort of Praia da Luz on May 3 last year shortly before her fourth birthday. No one has been arrested or charged in connection with her abduction.
The couple's counsel, Tim Scott QC, told the judge today there was no proof that Madeleine was alive, "but there is not a scrap of evidence that she is not".
He said the McCanns acknowledged the enormous effort devoted to the search by Leicestershire Police and other law enforcement agencies.
Shortly after their daughter disappeared, the judge, in the High Court Family division, granted them a wide order requiring anyone on whom it was served to disclose relevant information to their solicitors.
The police were among those served with the order and serious issues then arose about the public interest in maintaining confidentiality of police investigations.
The Serious Organised Crimes Agency and the Attorney General intervened to advance their own arguments on public policy.
Mr Scott said it became clear that areas of law of great interest to lawyers would have to be considered.
"Gerry and Kate McCann are not lawyers and so far as they were concerned the legal proceedings were moving further and further from the only matter which concerns them - the search for Madeleine.
"The proceedings were in danger of becoming a distraction from, rather than an aid to, that single goal," he said.
Mr Scott said the information to be released by the police related to telephone calls made to the McCanns' solicitors and passed on to Leicestershire Constabulary's incident room during the early stages of the inquiry.
The chief constable has now agreed to provide contact details and a summary of the information provided by those early callers.
The McCanns' search for their daughter continued in accordance with the title of the fund they set up to help them in their task, Madeleine's Fund - Leaving No Stone Unturned.
James Lewis QC, for the police, said Chief Constable Matt Baggott shared that aim, but felt it was vital to balance the understandable desire of the McCanns to have as much information as possible against the risks of compromising the continuing criminal investigation, damaging future international co-operation and breaching Portuguese law.
The material to be disclosed consisted of 81 pieces of information out of more than 11,000, said Mr Lewis.
Approving the settlement of the case, the judge amended her original wide disclosure order to state that it did not apply to Leicestershire Police or any other UK law enforcement agency.
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