The open letter, presented today in London, was signed by Oscar, Grammy and Nobel Prize winners, Olympic gold medallists, lords, knights of the realm, professors and novelists, representing "the best of British talent and intellect", according to the organisers.
Only people who live in Scotland get to vote in the September 18 referendum on whether to end the 307-year-old political union with England.
The famous signatories calling for a "No" vote -- Scots included -- do not have a vote but are hoping those who do keep the union together.
"Nevertheless, that decision will have a huge effect on all of us in the rest of the United Kingdom.
"We want to let you know how very much we value our bonds of citizenship with you, and to express our hope that you will vote to renew them.
"What unites us is much greater than what divides us. Let's stay together."
Hollywood stars included Dench, Patrick Stewart, Helena Bonham-Carter and Michael Douglas were on the list.
Besides Jagger, singing signatories include Bryan Ferry, World War II forces sweetheart Vera Lynn, Cliff Richard, Pink Floyd's David Gilmour, composer Andrew Lloyd Webber and Sting.
Olympic gold medallists including sailor Ben Ainslie, triple jumper Jonathan Edwards, rower Steve Redgrave, were among a host of sporting signatories.
Polls consistently put the "No" campaign ahead, though only by a handful of percentage points.
Prominent independence supporters include James Bond actor Sean Connery and "Trainspotting" novelist Irvine Welsh.
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