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The 90-year-old died "peacefully, surrounded by her family" in the early hours of Sunday at her home in Scotland, her family said.
Her daughter, actress Sophie Corbett, said: "We lost my darling Mum at one o'clock in the morning.
"She was an amazing woman, and my sister Emma and I, and all four of her grandchildren, Tom, Tilly, Dylan and Billy, will love and miss her forever.
"We will be bringing her home to Abingdon, near Croydon, which is where the family lived when my Dad was alive."
Hart was recently treated for a hip fracture in the Royal Infirmary of Edinburgh following a fall at the family estate Muirfield Wood in East Lothian.
She made her West End debut as a child in the Christmas show Where The Rainbow Ends, and also performed in child roles in operas at Covent Garden.
She found fame on stage in her 20s as the leading lady in Clown Jewels and Young In Heart with comedy troupe The Crazy Gang.
Hart met her future husband, the then up-and-coming comedian Ronnie Corbett, while working at Danny La Rue's club at Hanover Square in the late 1950s.
She was previously married to John Padley, a singer in The Four Jones Boys group, before she tied the knot with Corbett at Brixton registry office.
Hart played a show at Danny La Rue's club on their wedding night.
The couple's first child, Andrew, was born with a heart defect and lived for only six weeks.
Corbett and Hart went on to have two daughters and four grandchildren.
Corbett died aged 85 in 2016, having been diagnosed with a suspected form of motor neurone disease.
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Alongside Ronnie Barker, Corbett rose to fame in the popular BBC comedy sketch show The Two Ronnies.
The show, which pulled in up to 22 million viewers and was famous for its four candles sketch, ran for 16 years on the BBC.