The couple's son has been reportedly been called Deveraux Octavian Basil, after Elizabeth captioned a snap on her Instagram account: 'my beautiful baby brother Deveraux Octavian Basil Jagger' - although the picture had since been deleted.
A publicist for the 73-year-old Rolling Stones frontman confirmed the new arriva earlier this month in a statement which said that Melanie and Sir Mick were 'both delighted' with their new baby, while 'Mick was at the hospital for the arrival.'
The statement also said that: 'Mother and baby are doing well and we request that the media respect their privacy at this time.'
The baby is the first for Mick and Melanie, who have been in a relationship since 2014, but Mick has seven other children from previous relationships.
We first found out that the rockstar was to become a dad again in July. A spokesperson confirmed the news to The Sun, whilst a source close to the couple told the newspaper that the couple were 'surprised and happy' about the impending arrival.
'This is great news. It will be Mick's eighth child and nothing fazes him. He's been incredibly supportive,' they added.
'She's very bright, intelligent, sparky and has her own ideas about how she wants to lead her life. She knows what a great relationship Mick has with all of his children and can't wait for him to be a dad again.'
'They are taking the news in their stride.'
Mick first became a father in 1970, when he welcomed daughter Karis with singer and novellist Marsha Hunt. In 1971, a second daughter, Jade, was born to his then-wife Bianca Jagger.
By 1997, Mick had split from Bianca and begun his famous relationship with model Jerry Hall. They had four children together; Elizabeth, in 1984, James, in 1985, Georgia, in 1992, and Gabriel, in 1997.
Mick pictured with daughters Elizabeth and Georgia
He later welcomed another son with model and TV host Luciana Gimenez, named Lucas.
Mick is also grandfather to five children, and in 2014 became a great-grandfather when Jade's daughter Assisi gave birth to her daughter Ezra.
According to the source, Melanie will not move to London, where Mick currently resides, to raise the baby, and is instead expected to give up her dance career and bring up the child in America, either in Los Angeles or Connecticut, where she has family nearby.
'They've seen each other a lot, they like each other a lot but they still want independent lives,' they explained.
'Mick's relationship with Melanie works for both of them.'
'Melanie is very American and doesn't want to move to London, even though they've been seeing each other regularly over a long period of time now.'
'Mick likes the arrangement they have and doesn't want to be in another formal relationship.'
'But the most important thing is how much he loves his children and he has great relations with them. He will support and love this child.'