Police were called to her home in Wrotham, Kent, yesterday where her body was discovered.
Bob Geldof said: "Peaches has died. We are beyond pain."
Ms Geldof, a mother of two, was a prolific tweeter and the final message that she sent the day before her death was a picture of her as a child with her mother with the message "Me and my mum".
Ms Geldof was twice married and her second husband is musician Tom Cohen.
Bob Geldof said: "Peaches has died. We are beyond pain.
"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.
Ms Geldof's husband Tom 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 forever."
Ms Geldof was a young girl when her mother died tragically from an overdose in 2000 after splitting with Bob Geldof and forming a relationship with INXS frontman Michael Hutchence. Hutchence died in 1997.
Kent Police said the death was not being treated as suspicious.
A spokesman said: "Police were called at 1.35pm on 7 April 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 local coroner has been informed and a post-mortem examination is expected to be carried out in the next few days."
Ms Geldof had two young sons, 23-month-old Astala and 11-month-old Phaedra who was born on what would have been her mother Paula Yates' 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 amicably decided to split in February 2009 before divorcing in 2011.
Ms Geldof married Cohen, lead singer of south east London band S.C.U.M, in September 2012 at the same church in Davington, Kent, where her parents married 26 years earlier. It was also where her mother's funeral was held in 2000.
A flood of tributes was paid on social networking websites last night. 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".
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."
This Morning host Phillip 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."