Her Majesty said that she is “entirely supportive” of the decision by the duke and duchess to step back as “senior royals”, and said that the family had “constructive discussions” in a crunch meeting on Monday.
The Sussexes will spend time in Canada and the UK as the arrangements are thrashed out.
The statement read: “Today my family had very constructive discussions on the future of my grandson and his family.
“My family and I are entirely supportive of Harry and Meghan’s desire to create a new life as a young family. Although we would have preferred them to remain full-time working Members of the Royal Family, we respect and understand their wish to live a more independent life as a family while remaining a valued part of my family.
“Harry and Meghan have made clear that they do not want to be reliant on public funds in their new lives.
“It has therefore been agreed that there will be a period of transition in which the Sussexes will spend time in Canada and the UK.
“These are complex matters for my family to resolve, and there is some more work to be done, but I have asked for final decisions to be reached in the coming days.”
The Queen was joined at Sandringham by the Prince of Wales and Duke of Cambridge for the face-to-face talks with Harry.
It is unusual for the Queen to issue a statement in her own name – when not paying condolences after the death of a foreign head of state or sympathising following a natural disaster or terrorist attack.
The formal titles of the couple – the Duke and Duchess of Sussex – were not used by the monarch.
Earlier, William and Harry issued a joint statement denying a newspaper claim about the brothers’ relationship which they branded “offensive and potentially harmful”.
The statement did not name the newspaper but the Times has a front page story about the crisis, and says a source told the publication that Harry and Meghan “regarded themselves as having been pushed away by what they saw as a bullying attitude from the Duke of Cambridge”.
The report goes on to say that these allegations have been contested by other sources close to the Cambridges and Harry.
A joint statement issued on behalf of the princes said: “Despite clear denials, a false story ran in a UK newspaper today speculating about the relationship between the Duke of Sussex and the Duke of Cambridge.
“For brothers who care so deeply about the issues surrounding mental health, the use of inflammatory language in this way is offensive and potentially harmful.”
There has been much speculation about the relationship of the brothers who, a few years ago, were working closely together on charitable projects and lived yards from each other at Kensington Palace.
But Harry has conceded that the siblings are moving in different directions and their relationship, although close, has come under real strain.