Rev Richard Coles lived with David, 42, in their vicarage in Finedon.
He said in a statement on Twitter that David had died after a long illness and thanked Kettering General Hospital for their care.
Rev Richard Coles said: "I'm very sorry to say that @RevDavidColes has died. He had been ill for a while.
Kettering town centre restaurant ready to reopen with new menu and 'surprises'
Remembering the drought of 1976 across Northamptonshire in pictures from the archive
Prisons ombudsman concludes investigation after Higham Ferrers sex offender's death
Eight pumps tackling large fire at Corby recycling centre
Muslim woman left in tears after Kettering McDonald's wrongly puts bacon in burger
"Thanks to the brilliant teams who looked after him at @KettGeneral.
"Funeral details to follow. "The Lord shall be thine everlasting light, and the days of thy mourning shall be ended"."
In a reply, KGH said: "So sorry to hear this sad news. Please accept the condolences from everyone at Kettering General Hospital."
Before becoming a vicar, Mr Coles was a keyboard player in the 1980s band The Communards and is now a well-known broadcaster.
Rev Richard Coles appeared on Strictly Come Dancing in 2017 and co-presents Saturday Live on BBC Radio 4.
His dance partner on Strictly, Dianne Buswell, wrote on Twitter: "I'm so sorry to hear this. Sending all my love to you rev! My prayers and thoughts are with you x."
Others have been sending well-wishes to Mr Coles, including TV chef Nigella Lawson, who wrote: "I'm so very sorry for your loss and pain. Sending love."