Kate gave birth to her daughter at 8.34 this morning. She weighed 8lbs 3oz.
Her Royal Highness and her child are both doing well and the Duke of Cambridge was present for the birth.
Archbishop Justin Welby said: "It's wonderful news that The Duke and Duchess of Cambridge have a healthy new baby girl.
"Like many here and abroad, I'm delighted for the Royal couple on this special day.
"May God bless them and both of their children with love, health and joy."
Kensington Palace said in a statement: "Her Royal Highness The Duchess of Cambridge was safely delivered of a daughter at 8.34am. The baby weighs 8lbs 3oz.
"The Duke of Cambridge was present for the birth.
"The Queen, The Duke of Edinburgh, The Prince of Wales, The Duchess of Cornwall, Prince Harry and members of both families have been informed and are delighted with the news.
"Her Royal Highness and her child are both doing well."
To mark the occasion the Westminster Abbey Company of Ringers will ring a full peal of Cambridge Surprise Royal on the Abbey's ten bells at 2pm to coincide with the gun salute in Hyde Park.
It is a historic tradition that the Abbey's bells are rung for significant royal occasions.
Delighted that the Duke and Duchess of Cambridge have a healthy new baby girl. Wishing every blessing and happiness.— Justin Welby ن (@JustinWelby) May 2, 2015
The Dean of Westminster, the Very Revd Dr John Hall who conducted the wedding service of the Duke and Duchess of Cambridge in April 2011, said: "We assure The Duke and Duchess and their growing family of our prayers and very best wishes.
"As with the birth of Prince George, this birth will bring joy to people not just here but around the world, reminding us all of the delight we experience at a new birth in our own families."
The new-born is fourth in line to the throne.
It's thought the Christening of new Princess of Cambridge is likely to be this summer before the Queen takes her Balmoral break.
Premier's Antony Bushfield speaking to Shirley Tart, Church warden and the Royal Editor at the Shropshire Star:
Christians across the UK have been sending their best wishes to the couple.
Yippee!Such good news!Congratulations:William&Kate! May the Good and Generous Lord and Saviour, Jesus Christ,bless the Royal baby girl.Amen.— John Sentamu (@JohnSentamu) May 2, 2015
The Diocese of Lincoln would like congratulate to the Duke and Duchess of Cambridge on the birth of their baby girl. #RoyalBaby— Diocese of Lincoln (@CofELincoln) May 2, 2015
Congratulations to Prince William & Kate on the baby girl! #RoyalBaby— Hillsong London (@HillsongLondon) May 2, 2015
Really fantastic news that the Duke & Duchess of Cambridge have a healthy baby girl. Sending lots of prayers & thoughts their way today!— Vicky Beeching (@vickybeeching) May 2, 2015
Welcome to the world Baby Princess #RoyalBaby— OurChurchofEngland (@OurCofE) May 2, 2015
Congratulations to the Royal couple on the birth of their daughter from the Diocese of Truro— Diocese of Truro (@DioTruro) May 2, 2015