Judges: Time Richard was laid to rest
Richard was killed at the Battle of Bosworth in 1485 - ending the Wars of the Roses and the Plantagenet dynasty.
His body was taken to Leicester by supporters of the victorious Henry VII and buried in Greyfriars church, now the site of the council car park.
The alliance indicated it wanted the remains to be buried at York Minster, claiming that was the wish "of the last medieval king of England", who was known as Richard of York.
But their counsel Gerard Clarke told the court at a recent hearing the alliance would be satisfied if a consultation exercise was launched, and suggested the Queen and royal household should be at the top of the list of consultees.
Mr Clarke said it should also include the distant relatives themselves as well as members of the public.
He said the issue was important as the last English king to die in battle "is not just any old bones".
But Lady Justice Hallett, sitting with Mr Justice Ouseley and Mr Justice Haddon-Cave, ruled there were no public law grounds for interfering with the plans for reburial at Leicester Cathedral.
The three judges said in a joint ruling: "Since Richard III's exhumation on 5th September 2012, passions have been roused and much ink has been spilt.
"Issues relating to his life and death and place of re-interment have been exhaustively examined and debated.
"The Very Reverend David Monteith, the Dean of Leicester Cathedral, has explained the considerable efforts and expenditure invested by the cathedral in order to create a lasting burial place "as befits an anointed King".
"We agree that it is time for Richard III to be given a dignified reburial, and finally laid to rest."