A spokeswoman for the prime minister and his partner said both mother and baby are “doing very well”.
“The prime minister and Ms Symonds are thrilled to announce the birth of a healthy baby boy at a London hospital earlier this morning. Both mother and baby are doing very well,” the spokeswoman said.
“The PM and Ms Symonds would like to thank the fantastic NHS maternity team.”
It is understood Johnson was present throughout the birth.
The prime minister’s father Stanley said he was “absolutely delighted” and “thrilled” by the birth of his grandson.
Johnson will take a short period of paternity leave later this year, a Downing Street spokesman has confirmed.
The spokesman told a Westminster briefing: “I do expect the Prime Minister to take a short period of paternity leave later in the year, rather than now.”
Johnson returned to work in Number 10 following the birth, and was due to speak to Labour leader Sir Keir Starmer on Wednesday afternoon.
The new family are planning to live in their Downing Street flat along with their dog Dilyn, the spokesman confirmed.
Downing Street declined to say whether the baby was born prematurely, and the spokesman was unable to provide details of the weight, timing, nature or location of the birth.
Fellow politicians were quick to tweet their congratulations to the PM and his partner.
Chancellor Rishi Sunak tweeted that it was “great to hear that Downing Street is getting a new resident”.
Health secretary Matt Hancock, meanwhile, expressed his “unalloyed joy” at the announcement.
Labour leader Keir Starmer congratulated the couple on “wonderful news”, and Scottish first minister Nicola Sturgeon wished “health and happiness to the wee one”.
Johnson, 55, and Symonds, 32, announced in March that they were expecting a baby in “early summer”, and that they had become engaged at the end of last year.
The news of the birth comes just weeks after the PM was discharged from intensive care, where he had been battling coronavirus.
So thrilled for Boris and Carrie. Wonderful to have a moment of unalloyed joy!— Matt Hancock (@MattHancock) April 29, 2020
Some good news - sending congratulations to Carrie and the PM. And wishing health and happiness to the wee one. https://t.co/V2k1KKqVN7— Nicola Sturgeon (@NicolaSturgeon) April 29, 2020
He returned to work on Monday to take charge of the UK’s coronavirus crisis after recuperating from his own illness at Chequers, the prime minister’s country retreat.
Symonds also suffered symptoms of the disease, and described her experience of the illness while pregnant as “obviously worrying”.
She said on 4 April: “I’ve spent the past week in bed with the main symptoms of coronavirus. I haven’t needed to be tested and, after seven days of rest, I feel stronger and I’m on the mend.”
The PM has previously suggested he intends to take paternity leave, although it is not known if this remains the case given the coronavirus crisis.
Johnson has four children with his second wife, lawyer Marina Wheeler – Theodore, Cassia, Milo and Lara. The couple’s marriage ended in September 2018 after 25 years together.
In 2013, the Appeal Court ruled that the public had a right to know that Johnson had fathered a daughter during an adulterous liaison while mayor of London in 2009.