The couple will also no longer formally represent The Queen.
The Duke and Duchess of Sussex intend to repay £2.4m of taxpayer money for the refurbishment of Frogmore Cottage, which will remain their UK family home, the statement added.
The new arrangement comes into effect in spring this year, the palace said.
The statement comes after senior royals held talks on Monday about the future role for the couple, who this month announced they wanted to "step back" as senior royals.
The Queen said following "many months of conversations and more recent discussions" she was "pleased that together we have found a constructive and supportive way forward for my grandson and his family".
"Harry, Meghan and Archie will always be much loved members of my family," the statement continued.
"I recognise the challenges they have experienced as a result of intense scrutiny over the last two years and support their wish for a more independent life.
"I want to thank them for all their dedicated work across this country, the Commonwealth and beyond, and am particularly proud of how Meghan has so quickly become one of the family.
"It is my whole family's hope that today's agreement allows them to start building a happy and peaceful new life."
Buckingham Palace said the royal couple understood they were required to step back from royal duties, including official military appointments.
The palace added that they would continue to maintain their private patronages and associations.
"While they can no longer formally represent the Queen, the Sussexes have made clear that everything they do will continue to uphold the values of Her Majesty," the statement added.
"The Sussexes will not use their HRH titles as they are no longer working members of the Royal Family."
HRH, an abbreviation of His/Her Royal Highness, is used as part of the title of some members of the royal family, including prince or princess.
The palace said it would not comment on what security arrangements would be in place for the royal couple.
Their new website, sussexroyal.com, has been updated following the Queen's statement.
"In line with the statement by Her Majesty The Queen, information on the roles and work of the Duke and Duchess of Sussex will be updated on this website in due course," it said.
The announcement marks the conclusion of talks about the couple's future with senior members of the family and royal aides.
Earlier this month, they said they wanted a "progressive new role" within the institution, where they would be financially independent and divide their time between the UK and North America.
Last year, they both spoke about the difficulties of royal life and media attention.
The duke said he feared his wife would fall victim to "the same powerful forces" that led to his mother's death.