British Vogue editor Edward Enninful also offered his congratulations, writing: "Seeing the two of you together made my year. You will bring each other so much joy."
Ms Campbell's mother, Valerie Morris Campbell, posted on Instagram: "I'm beyond thrilled as I've waited a long time to be a grandmother."
No further details about the new baby have been made available, but during a 2017 interview with the Evening Standard, Ms Campbell spoke about wanting to become a mother.
She was quoted as saying: "I think about having children all the time. But now with the way science is I think I can do it when I want."
The catwalk star was discovered as a schoolgirl and went on to become the first black British model to appear on the cover of British Vogue.
She is also known for her charity work, and in 2005 founded Fashion For Relief, a charity merging fashion and philanthropy.