"Florence becomes the second child born to a serving leader in a decade after Leo Blair in 2000, who was the first legitimate child born to a serving prime minister in over 150 years — since the birth of Francis Russell in July 1849." states the Guardian today.
This is the second time I've seen that proviso legitimate, which strikes me as unusually cautious for the modern British journalist who is usually cavalier about historical accuracy. So, have there been any illegitimate children born to serving Prime Ministers? A quick skim through the Wikipedia biographies doesn't reveal any. I've only found two bastards, but neither was born, or even conceived, when their father was in office. Lord Palmerston was reputed to be the father of Lady Emily Cowper, whose mother later became his wife. The second, of course, is Lloyd-George's daughter by Frances Stevenson.