David Cameron's official spokesman described the announcement as "quite terrific news", telling reporters: "The Prime Minister will send his very, very best wishes to the happy couple."
Zara Phillips and Mike Tindall have announced they are expecting their first baby in the new year.
The birth will follow the keenly anticipated arrival of the Duke and Duchess of Cambridge's baby, which is expected in the coming days.
Buckingham Palace, using Zara's married name, said in short statement: "Mr and Mrs Mike Tindall are very pleased to announce that Zara Tindall is expecting a baby in the new year.
"The Princess Royal and Captain Mark Phillips, Mr Phillip and Mrs Linda Tindall, and members of both families are delighted with the news.
"This baby will be the first child for Mike and Zara and the third grandchild for the Princess Royal."
Zara Phillips and Mike Tindall are expecting a baby in the New Year, Buckingham Palace announced today.
Zara Phillip retired from competing at the Badminton Horse Trials after making a mistake on the cross-country course.
After a good start during yesterday's dressage stage, Phillips - who was riding High Kingdom - missed the second of two brush jumps in the Lake and pulled up.
It means she can no longer take part in tomorrow's show jumping stage.
Zara Phillips put herself firmly in contention at the Mitsubishi Motors Badminton Horse Trials today with a quality dressage performance on her London 2012 horse High Kingdom.
Phillips, the Queen's granddaughter, posted a dressage score of 44.3 penalties that kept her firmly in touch with the leading group midway day two.
Braving the cold at the Cheltenham Festival the coats have stayed on and lots of fur has been on show, but despite the weather the hats have been no less impressive.
Styles this year seem to favour smaller hats with plenty of feathers making an appearance.
How does Ladies' Day compare to last year?
Zara Phillips is putting her equestrian skills to a new use - by perfecting the Gangnam Style dance.
Phillips, the granddaughter of the Queen, will be joined by other figures from the Olympic eventing team for the performance in aid of Children In Need.
Zara Phillips has told Daybreak that she is hungry for more success after becoming the first British royal to win an Olympic medal.
Joined by the rest of the Team GB eventing team, she told Daybreak: "We'll take it one year at the time, in our sport it's better to do just yearly goals but being at one Olympics make you hungry for the next one."
Zara Phillips delivered a good performance on her Olympics cross-country debut at Greenwich Park to keep Great Britain firmly in medal condition.
The Queen's granddaughter was watched from the main arena stands by the Duke and Duchess of Cambridge, Prince Harry and the Duchess of Cornwall.
The 31-year-old former world champion produced a clear round inside the time, completing a course that claimed its fair share of casualties.
Seven of the first 27 combinations fell and were eliminated, but Phillips kept control all the way around a testing track that was very slippery in places.
Five combinations were clear inside the time as the competition entered its second half - Phillips, Wilson, Germany's Ingrid Klimke, Sweden's Sara Algotsson Ostholt and Australian Christopher Burton.
Phillips finished on her dressage score of 46.10.
Zara Phillips been chosen alongside William Fox-Pitt, Mary King, Piggy French and Tina Cook for London 2012.
She clinched her spot in the team following a third-placed finish at the Bramham International CIC three-star class after posting a personal best dressage score, and then jumping clear in the showjumping and cross-country phases.