Matthew, 53, admitted the couple wept "tears of joy" after the miracle birth, following eight years of trying to start a family.
Amelia, 42, cradled Cassady and Lorraine Kelly cooed over the tot, who was conceived after six IVF attempts.
Amelia said: "To finally have her and hold her in my arms after all that time trying… it's like a dream."
Matthew added: “We are weeping tears of joy, I’m so happy…it’s the strongest emotional connection. It is a bubble at the moment, all you do is eat sleep and baby.
"It is a journey and a rollercoaster, with some huge lows and some giddy highs. One of the loveliest things, social media…to have the outpouring of love from people.”
Fans gushed over the couple online and one wrote on Twitter: "Lovely to see Mathew And Amelia Wright with their gorgeous baby they both look completely besotted with her,cherish every moment because they grow so fast."
Another added: "What a lovely couple, Matthew Wright and wife are. Beautiful baby Cassidy. Enjoy"
Amelia explained baby Cassady had to stay in a special care unit to begin with in hospital as she was born prematurely.
She said: "It was devastating seeing her hooked up to the machines. I had the C section and then I sat there for an hour staring at her, it was very emotional, but she bounced back quickly."
The couple married in 2010, after Amelia had an ectopic pregnancy in 2009 having fallen pregnant with twins.
Their seventh attempt saw them have to decide whether to donate their last embryo to medical science or to try pregnancy one more time.
Back in September, Matthew and Amelia appeared on Lorraine and she explained: "We had an embryo in the freezer for six years.
"We were talking about donating it to medical science because I wasn't sure if I could go through another round of heartbreak and disappointment.
"When it came down to it, I couldn't give it up. I had to just give it one last shot."
After being played a montage of clips from celebrity pals sending their love and support to the couple, Matthew said he was "completely overwhelmed".
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