The rescue dog came from the Friends of Animals Wales charity, which said the "gorgeous pup will be living his best life with Carrie and the Prime Minister at their Downing Street home".
The Prime Minister's official spokesman said: "They are a fantastic charity run by volunteers who work around the clock to make a difference to animals.
"The Prime Minister has always been a passionate supporter of animal welfare and has always believed that animals should always get the right start in life.
"That's why the Government has taken such significant action in this area, and that obviously includes introducing Lucy's Law."
Lucy's Law was aimed at cracking down on puppy farms and the new Downing Street dog started life in one such establishment.
In a statement, Friends of Animals Wales said: "We knew that Carrie Symonds has always been a huge animal welfare advocate and we are absolutely overjoyed that this gorgeous pup will be living his best life with Carrie and the Prime Minister at their Downing Street home.
"To think our little 'wonky' pup started his life in a Welsh puppy farm and was destined for an uncertain existence at the hands of a puppy dealer, but is now going to be with people who absolutely adore him is a dream come true.
"This little man will want for nothing in his life, what could be better than that?
"And how fantastic to be raising the profile of rescue dogs everywhere in the process."
The PM's spokesman could not confirm whether the puppy - who is a few months old - had been toilet trained.
"It's a rescue puppy, it has been rescued by the charity Friends of Animals Wales, and they have been caring for the puppy so far," the spokesman said.
The dog belongs to Mr Johnson and Ms Symonds and would be expected to leave Number 10 when the PM leaves office.
Larry the Number 10 cat was expected to meet the puppy on Monday morning.