Bob Geldof said his family were “beyond pain” after his daughter Peaches was found dead at the age of 25.
The mother-of-two, whose mother was the late TV presenter Paula Yates, was found at her home in Wrotham, Kent, yesterday.
Ms Geldof was a prolific tweeter and the final message she sent on Sunday was a picture of her as a child with her mother, with the message “Me and my mum”.
Ms Geldof had two young sons with her second husband, musician Tom Cohen.
Bob Geldof said: “She was the wildest, funniest, cleverest, wittiest and the most bonkers of all of us.
“Writing ‘was’ destroys me afresh. What a beautiful child. How is this possible that we will not see her again? How is that bearable?
“We loved her and will cherish her forever. How sad that sentence is.
“Tom and her sons Astala and Phaedra will always belong in our family, fractured so often, but never broken.”
The statement was signed Bob, Jeanne, Fifi, Pixie and Tiger Geldof.
Cohen said: “My beloved wife Peaches was adored by myself and her two sons Astala and Phaedra and I shall bring them up with their mother in their hearts every day. We shall love her for ever.”
Ms Geldof was a young girl when her mother died from an accidental heroin overdose in 2000, aged 40.
Yates was divorced from Bob Geldof in 1996 after forming a relationship with INXS frontman Michael Hutchence.
Hutchence was found dead in a hotel room in Sydney, Australia, in 1997, and Yates went on to lose custody of the three daughters she had with Geldof – Peaches, Pixie and Fifi – the following year.
Bob Geldof later adopted Yates and Hutchence’s daughter Heavenly Hiraani Tiger Lily.
Ms Geldof’s oldest son Astala was 23 months old, while 11-month-old Phaedra was born on what would have been her mother’s 54th birthday.
She was 19 when she married US musician Max Drummey at the Little White Wedding Chapel in Las Vegas in August 2008. The couple announced that they had decided to split amicably in February 2009 before divorcing in 2011.
Ms Geldof married Cohen, lead singer of south-east London band SCUM, in September 2012 at the church in Davington, Kent, where her parents married 26 years earlier. It was also where her mother’s funeral was held.
Kent Police said: “Police were called at 1.35pm on April 7, 2014 to an address near Wrotham following a report of concern for the welfare of a woman. A woman aged 25 was pronounced dead by South East Coast Ambulance Service.
“The death is being treated as a non-suspicious, but unexplained, sudden death.The local coroner has been informed and a post-mortem is expected to be carried out in the next few days.”
Among the people paying tribute online were This Morning hosts Phillip Schofield and Holly Willoughby.
Schofield, who appeared with Ms Geldof on the ITV show, wrote: “Utterly stunned at the terrible news of the death of Peaches Geldof! The dreadful loss of a really lovely woman.”
Willoughby said: “Just heard about Peaches, so unbelievably tragic. So shocked and sad. Sending my thoughts and love to her family.”
Singer Ellie Goulding said: “Even if you think you’ve got it all figured out, some things still can’t be explained or understood. Two beautiful children. RIP Peaches.”
Model Daisy Lowe posted a picture of a broken heart on Twitter.
Lily Allen said: “My thoughts are with Peaches’ family at this awful time. I hope they get to grieve in peace. Peaches, rest in peace gorgeous girl.”
Mother & Baby magazine announced in February that Ms Geldof was to be their new columnist.
A spokesman said it was after she had completed what they called, a “mama metamorphosis” from party girl to working mother. It was to feature her attempts to juggle parenting, work and family.
A tribute posted online by the magazine reads: “We are extremely saddened at the tragic news that Peaches Geldof has died. Our hearts and sympathies are with her family and two little boys.”