Earlier this month, the Duke tweeted a personal message to the Prime Minister and Carrie as Mr Johnson battled coronavirus in hospital.
"I couldn't be happier".
Ms Symonds said the child was named after Mr Johnson's grandfather Wilfred and her grandfather Lawrie.
Wilfred Johnson was born healthy at University College Hospital in London Wednesday morning, she said, adding that the couple wanted to thank their "fantastic" maternity team.
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In a post on Instagram, Carrie Symonds announced that she and Boris Johnson have chose to name their newborn child Wilfred Lawrie Nicholas Johnson. "My heart is full", Symonds wrote alongside a picture of her cradling her new son.
He spent the first 10 days of his illness isolating in Downing Street but after his symptoms worsened, he was rushed off to the hospital on April 5.
The medics shared their "warm congratulations" with the couple, and said in a statement: "We are honoured and humbled to have been recognised in this way, and we give our thanks to the incredible team of professionals who we work with at Guy's at St Thomas' and who ensure every patient receives the best care".
They announced in February they were expecting their first child and that they were engaged to be married.
Johnson was expected to take statutory paternity leave at some point in the future, his spokesman told reporters this week.
The new arrival is the third baby born to a serving prime minister in recent history. The wives of Tony Blair and David Cameron had children during the tenures of their husbands.