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It is believed he contracted the virus in April, around the same time that his father Prince Charles also caught it.
While the palace did not comment officially, according to The Sun - which first reported the story - William, 38, kept his diagnosis private to avoid alarming the public.
The newspaper cited William as having told 'one observer at an engagement': "There were important things going on and I didn't want to worry anyone."
A source also told The Sun: "William was hit pretty hard by the virus - it really knocked him for six. At one stage he was struggling to breathe, so obviously everyone around him was pretty panicked."
William was treated by palace doctors and followed government advice by isolating at home at Anmer Hall in Norfolk.
He carried out 14 telephone and video call engagements during April.
Jonny Dymond, BBC royal correspondent, also said William's condition may not have been public knowledge at the time in a bid to avoid further alarm, given the mood of the nation at the time.
"But the palace also tries to preserve some privacy for the Royal Family," Dymond added.
Prince Charles, 71, tested positive for the virus in March.
He had shown some symptoms of coronavirus, but thankfully these were only mild and otherwise the royal was not in poor health.
His wife Camilla, the Duchess of Cornwall, was also tested for the disease, but her test showed as negative.
A statement released by Clarence House reads: "The Prince of Wales has tested positive for Coronavirus. He has been displaying mild symptoms but otherwise remains in good health and has been working from home throughout the last few days as usual.
"The Duchess of Cornwall has also been tested but does not have the virus. In accordance with Government and medical advice, the Prince and Duchess are now self-isolating at home in Scotland.
"The Test were carried out by the NHS in Aberdeenshire where they met the criteria required for testing.
"It is not possible to ascertain from whom the Prince caught the virus owing to the high number of engagements he carried out in his public role during recent weeks."
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