She will be stepping down from the Radio 4 show for a “number of months” from the end of September.
Fellow broadcaster Lauren Laverne, 40, is to host the long-running show while Kirsty is away.
Fibromyalgia, also called fibromyalgia syndrome or FMS, is a long-term condition that causes pain all over the body, according to the NHS.
In a statement issued by the BBC, Kirsty said: “Casting away some of the world’s most fascinating people is a wonderful job.
“However, I’m having to take some time away from Desert Island Discs as I’m suffering from a form of fibromyalgia.
“I wish Lauren all the very best. I know she will be great and I very much look forward to getting back to good health and back to work.”
Lauren said: “It is a huge honour to be asked to cover my favourite programme on behalf of one of my heroes.
“As a listener first and foremost I know what a special place Desert Island Discs holds in the hearts of the public.
“So much of that is down to the warmth, wit and peerless skill of Kirsty Young.
“I wish Kirsty better and will do my very best to look after Desert Island Discs until her return.”
A new series of the show is scheduled to start on September 16 and Kirsty will host the first two episodes with castaways opera singer Danielle de Niese and neurosurgeon Henry Marsh.
Former Five News presenter Kirsty recently won praise for her TV coverage of the Duke and Duchess of Sussex’s wedding.
She took over from Sue Lawley as host of Desert Island Discs in 2006. Laverne will host the third episode of Desert Island Discs onwards and will continue to present her weekday mid-morning show on BBC Radio 6 Music.
She becomes 6 Music’s new weekday Breakfast Show presenter in January next year.