The new arrival, whose name has not yet been announced, is the first child for 21-year-old Assisi and her chef boyfriend Alex Key.
Her mother Jade, 42, who is expecting a baby boy next month, is now a grandmother.
Announcing her pregnancy to HELLO! magazine, the young mum said that early motherhood runs in the family; Jade was just 20 when Assisi was born to her and former partner artist Piers Jackson.
"The women in our family just tend to have children young, it runs in the genes," said Assisi, who lives in Cornwall with her 25-year-old boyfriend.
"I grew up quite quickly and wouldn't have considered having a child if I was doing what my friends are doing at this stage, messing around," added Assisi. "I don't drink, I don't smoke and I'm very settled."
Jade paid tribute to her daughter's nurturing skills. "She'll be a perfect mum," said Jade. "Assisi is a level-headed, sweet girl and I think it will give them both a lot of pleasure. It gives you an amazing purpose in life, to fulfil nature's destiny."
Assisi's daughter is unlikely to call Sir Mick Great-Grandpa. Talking about the rocker's reaction to her pregnancy news, she said: " I imagine it's nice to be a great-grandad – although I'm not sure he likes the idea of getting old, or being called one.
"I call him Mick – I wouldn't start calling him Grandpa."