Video report by ITV News Political Correspondent Carl Dinnen
Both mother and baby are doing very well, and the Prime Minister returned to work in Number 10 following the birth a Downing Street spokesperson said.
The spokesperson said: "The Prime Minister and Ms Symonds are thrilled to announce the birth of a healthy baby boy at a London hospital earlier this morning.
"The PM and Ms Symonds would like to thank the fantastic NHS maternity team."
Mr Johnson is thought to have been present throughout the labour, sources say.
It was announced in February that the couple were expecting their first child together and had got engaged.
A spokesperson at the time had said the baby was due in "the early summer" suggesting the new baby boy arrived prematurely.
On Wednesday afternoon a spokesman for Number 10 declined to confirm whether the baby was born prematurely, and was unable to provide details of the weight, timing, nature or location of the birth.
Earlier in April, Mr Johnson was taken into critical care for treatment contracted coronavirus in the outbreak, the 55-year-old required "oxygen support".
Ms Symonds also suffered symptoms of Covid-19 at the beginning of the month - as a pregnant woman at the time, the 32-year-old would have fallen into the group of people deemed vulnerable in the pandemic.
ITV News Political Correspondent Carl Dinnen reports on whether the PM is likely to take paternity leave:
The Prime Minister only returned to Downing Street on Monday, after taking time to recover from coronavirus at Chequers.
On whether Mr Johnson will take his paternity leave, a Downing Street spokesman told a Westminster briefing: "I do expect the PM to take a short period of paternity leave later in the year rather than now".
The prime minister had previously suggested he did intend to take paternity leave but this was before the coronavirus outbreak and subsequent lockdown.
The PM's father Stanley Johnson told ITV News he is "absolutely delighted" at the birth of his grandson.
He added: "It's wonderful news - I don’t suppose he’ll be taking PMQs today but he's got a pretty sound reason".
Boris Johnson did not be attend Prime Minister's Questions on Wednesday with Foreign Secretary Dominic Raab standing in instead.
Mr Raab told the House: "The PM and his fiancee Carrie Symonds have announced the birth of a health baby boy this morning, both mother and baby are doing well and I’m sure the whole House will want to join with me in sending congratulations and our very best wishes to them."
The Foreign Secretary opened Wednesday's government daily coronavirus briefing by saying he had spoken to Mr Johnson and that "both mum and son are doing really well".
The Queen has sent a private message of good wishes to the couple to congratulate them on the birth of their son, Buckingham Palace said on Wednesday afternoon.
Congratulations have been flooding in for the couple from colleagues and across the political spectrum.
Former PM David Cameron tweeted his congratulations to the couple, writing: "Sam and I are thrilled for you both! Sorry we didn’t leave the cot - but the climbing frame should still be in the garden!"
Health Secretary for England Matt Hancock tweeted: "So thrilled for Boris and Carrie. Wonderful to have a moment of unalloyed joy!"
Chancellor Rishi Sunak wrote: "Great to hear Downing Street is getting a new resident" and wished the couple "congratulations".
Speaker Lindsay Hoyle said: "Such happy news amid so much uncertainty - 2020 is certainly a year they will never forget.
"On behalf of everyone in the House of Commons, may I say congratulations to the Prime Minister and Carrie Symonds on the birth of their son."
First Minister for Northern Ireland, Arlene Foster wished congratulations to the couple, writing on Twitter: "Wonderful news of the birth of a baby boy! [...] More sleepless nights ahead!"
While Nicola Sturgeon, First Minister for Scotland, wished "health and happiness to the wee one."
Australian prime minister Scott Morrison also tweeted congratulations to the new parents, as did Japan's prime minister Shinzo Abe.
'It's a really difficult time' for new parents doctor tells ITV News:
Elsewhere Sir Ed Davey - acting leader of the Liberal Democrats - tweeted: "Many congratulations to the PM & Carrie Symonds on the birth of their son!"
While Sir Keir Starmer wrote: "Wonderful news. Many congratulations to Boris Johnson and Carrie Symonds."
Addressing the House of Commons at PMQ's on Wednesday, the Labour leader said: "Whatever differences we have in this House, Mr Speaker, as human beings I think we all recognise the anxiety that the Prime Minister and Carrie must have gone through in the past few weeks.
"I really hope that this brings them incredible relief and joy".
Boris Johnson spoke with Sir Keir on the phone on Wednesday afternoon in what Downing Street said was a "constructive" call - the pair have agreed to continue speaking about the national effort to defeat coronavirus.
Mr Johnson, 55, and Miss Symonds, 32, made history as the first unmarried couple to officially live together in Downing Street when they moved in last year.
In recent years Number 10 has been home to a number of babies, with Tony Blair and David Cameron both adding to their families while serving as PM.
A name for the newborn is yet to be announced, but Wilfred is currently the bookies' favourite.
Alexander and Winston are also thought to be likely options considered by the couple - but Wilfred was proving so popular with punters on Wednesday morning that Ladbrokes slashed odds to just 3/1.
Mr Johnson is thought to be a father to five other children.