Donald Trump Jr, 42, was diagnosed at the start of this week and has been quarantining at his hunting cabin since the result, the spokesman said.
"He's been completely asymptomatic so far and is following all medically recommended Covid-19 guidelines," according to the statement.
Don Jr is the second of the president's children to test positive.
Barron Trump, 14, was also diagnosed last month but made a swift recovery.
A firebrand speaker, Don Jr played a major role in his father's presidential campaign.
There has also been speculation that Don Jr is interested in running for the White House, conjecture he hasn't tried to tamp down.
Don Jr's partner, Kimberly Guilfoyle, a former Fox News host, tested positive for the disease in July, and also recovered. He apparently did not contract the infection at the time.