A post-mortem examination proved inconclusive, prompting further investigations including toxicology tests to try to establish her cause of death.
Mother-of-two Ms Geldof, 25, was found at her home in Wrotham, Kent, on Monday and police have said her death is being treated as a "non-suspicious, unexplained sudden death".
A spokesman for North West Kent coroner Roger Hatch said: "We can confirm that the body has been released to the family for funeral arrangements to be made."
Ms Geldof's body was found by police after officers were called "following a report of concern for the welfare of a woman".
Her father Bob Geldof and other relatives led the tributes after her death.
A message signed by the Live Aid organiser, his partner Jeanne Marine, and Ms Geldof's sisters Fifi Trixibelle, Pixie and Tiger said the family was "beyond pain", adding: "She was the wildest, funniest, cleverest, wittiest and the most bonkers of all of us."
Ms Geldof was just 11 when her mother, TV presenter Paula Yates, died from an accidental heroin overdose in 2000, aged 41.
She married Tom Cohen, lead singer of London band Scum, in September 2012 and the couple have two sons, Astala, 23 months, and 11-month-old Phaedra.
In a column for Mother & Baby magazine, she wrote she was "happier than ever" after becoming a mother.