The six-year-old, who suffered from cerebral palsy and a rare and severe epilepsy syndrome, was taken ill overnight and died at St Mary's Hospital in Paddington, London.
Ivan, who required round-the-clock care all his life, was the eldest child of the Tory leader and his wife Samantha.
Prime Minister's Questions, which Mr Cameron had been due to attend at noon, was called off at Gordon Brown's suggestion.
The Prime Minister, whose own baby daughter died in 2002, said: "Sarah and I were very saddened to hear of the death of Ivan and we have sent our condolences to David and Samantha. The death of a child is a loss no parent should have to bear.
"I know Ivan was a child who brought joy to all those who knew him and his was a life surrounded by love. The thoughts and prayers of the whole country are with David, Samantha and their family."
Buckingham Palace said the Queen has sent a private message of sympathy to the Camerons.
A Conservative Party spokesman said: "It is with great sadness that David and Samantha Cameron must confirm the death of their six-year-old son Ivan. David and Samantha would ask that their privacy is respected at this terribly difficult time."
Mr and Mrs Cameron have two other children, Nancy, five, and Arthur, three.
It is understood the Camerons were with Ivan at the time of his death. Ivan was diagnosed with Ohtahara Syndrome - a very rare epilepsy syndrome that occurs in childhood.

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