The 39-year-old announced she had given birth to a baby boy on 20 June and the new mum revealed her choice of baby name on Twitter.
"Proud and excited to announce the arrival of my beautiful baby boy Roman," Barrett wrote alongside a string of excited emojis.
Fans were quick to praise Barrett for her choice.
The singer announced she was expecting her first child with her partner Paul Cashmore, just three months ago, in March 2016.
She previously said she was worried her age might affect her chances of falling pregnant.
“I’m 40 this year so there was a part of me that was worried I might have left it too late,” she told OK! Magazine at the time of announcing her pregnancy.
“But thankfully that wasn’t the case.”
On 14 June, Barrett updated fans on her pregnancy, telling them she had just "days to go".