In a statement released later by the second-time parents they revealed that they were 'totally smitten' with their newborn within minutes of her birth.
They said: "We're so happy that baby has arrived safely. Within minutes of meeting her we were totally smitten. "
And they're not the only ones – Amelie is over the moon at the prospect of having a little playmate. "Amelie is happy and ‘cited to have a little sister to play with and we can't wait to introduce her to the whole family," they said.
Last month the brunette singer posted a snap of her blooming baby bump on Twitter.
"BABY BUMP - I'm starting to nest as it's not long to go now! she wrote alongside the picture," she wrote.
Rachel, 35, shared her pregnancy news on her website, alongside a sweet picture of the family's shoes with their birth years chalked underneath.
Next to Amelie's pink trainers were a pair of white baby shoes with the year 2014 written in chalk below them.
Rachel and Alex have been married four and a half years. They tied the knot in 2009 in a lavish afternoon ceremony at Claridge's hotel in London's Mayfair.