Symonds (32) had said previously that their baby was due in the early summer. Johnson (55), whose country is facing its worst health crisis since the 1918 influenza outbreak, will take a short period of paternity leave later in the year.
"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 couple's spokeswoman said. "Both mother and baby are doing very well."
Johnson was present at the birth, the Prime Minister's spokesman said later. There was no announcement of a name. Queen Elizabeth has sent a private message of good wishes to the couple, Buckingham Palace said.
The new arrival tops a tumultuous month for Johnson. He returned to work on Monday after recuperating from Covid-19, which had left him seriously ill in intensive care at the peak of the coronavirus outbreak.
Symonds, a former public relations executive, also had symptoms of the virus but recovered more swiftly.
"This birth would be counted as premature," said Alexander Heazell, professor of obstetrics at the University of Manchester.
A review of SARS-1 and Covid cases suggested that the preterm birth rate was about double the normal rate, he added.
Johnson's Conservative government has faced criticism for its handling of the coronavirus epidemic and its slow initial response. The country now has one of the world's highest death tolls, with more than 21,000 fatalities.
But he benefited from public sympathy over his hospitalisation, and state broadcaster the BBC and other media outlets made the baby's birth the top item on their news bulletins.
"Great to hear Downing Street is getting a new resident," said Johnson's finance minister, Rishi Sunak.
Opposition Labour Party leader Sir Keir Starmer said: "Wonderful news. Many congratulations to Boris Johnson and Carrie Symonds."
England's chief midwife also joined in the congratulations.
"I am incredibly proud of the NHS maternity team who looked after them," said Jacqueline Dunkley-Bent.
Symonds and Johnson have been living together in Downing Street since he became Prime Minister in July last year. They announced in February they were expecting their first child and were engaged to be married.
Johnson, once dubbed "Bonking Boris" by Britain's tabloid media, has a complicated private life.
He was once sacked from the Conservative Party's policy team while in opposition for lying about an extra-marital affair. He has been divorced twice and refuses to say how many children he has fathered.
In 2013, when Johnson was mayor of London, appeal court judges ruled that the public had the right to know he had an extra-marital affair with a woman who gave birth to his daughter.
Johnson's previous marriage was to Marina Wheeler, a lawyer. They had four children together but announced in September 2018 that they had separated. They divorced earlier this year.
Few British leaders have had babies while in office, although it has become more common among recent occupants of Downing Street.
David Cameron's wife, Samantha, gave birth to their daughter, Florence, in 2010 and Tony Blair's wife Cherie gave birth to their son, Leo, in 2006.
Before that, Frances, the wife of John Russell, a 19th-Century Whig and Liberal politician, gave birth to two children in 1848 and 1849 while he was in office.
