The couple share their joy in a special world exclusive photoshoot and interview inside their Hollywood home.
The baby's "besotted" grandparents, Prince and Princess Michael of Kent, happily join in the shoot as they are on a week long holiday visiting their first grandchild little Maud Elizabeth Daphne Marina.
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Speaking for the first time since the baby was born on 15 August, Freddie said of choosing the name for their precious first child, who is 42nd in line to the throne: "We liked the idea of an old-fashioned English name and there are a couple of Princess Mauds in my father's family from a hundred years ago – one was the youngest daughter of Edward VII who ended up being Queen of Norway, and another a grand-daughter of Edward VII by his daughter Princess Louise.
"It's also a beautiful, elegant name and it was always our first choice. Elizabeth is a tribute to the Queen, Daphne is Sophie's grandmother, who she was very close to and Marina is my paternal grandmother."
Describing Maud's birth at Ronald Reagan UCLA Medical Centre in Los Angeles, 33-year-old actress Sophie tells HELLO!:
"It was long, hard and by the end I was begging for a caesarian thinking 'I can't do this anymore' but I managed a natural birth with some wonderful doctors and midwives along with Freddie in the room with me, who were just stunningly brilliant, saying, 'You can do this!'. Their dynamic, American positivity carried me through."
Freddie adds of his his wife of almost four years : "It was horrible to see Sophie in discomfort but I trusted the excellent doctors and nurses that she would get through it all right. Sophie did incredibly well, I was very proud."
Sophie said Freddie was the perfect birthing partner telling the magazine: "He was very brave, because it wasn't delicate or pretty and I was horrid to him. He read me a crossword clue in the middle of it and I just wanted to throw a saucepan at his head!"
Prince Michael told HELLO!: "We are absolutely thrilled by the arrival of our beautiful first grand-daughter, Maud. We are both besotted with her. Not only does she have the prettiest blue eyes and adorable tiny feet, but she is also remarkably well-behaved."
Speaking of this second new addition to the British Royal family this summer, Princess Michael added: "We haven't met Prince George yet as we have been on holiday – and of course we've been preoccupied with the birth of our granddaughter – but we look forward to doing so at Christmas."
Sophie says of Maud and Prince George meeting for the first time: "Perhaps we'll introduce the babies at Buckingham Palace for the Christmas lunch, that would be very sweet."
For the full photoshoot and interview see HELLO! magazine issue 1293, out now. There are extra photos in our iPad edition, available from the iTunes store by clicking here.