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.
Announcing her pregnancy, Assisi said that early motherhood runs in the family; Jade was 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.
Talking about her rocker grandfather's reaction to her pregnancy news, Assisi 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."