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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."