"Happiest day of my life my first son was born this morning xxx", she wrote on Monday.
The model turned entrepreneur has welcomed her baby, who was born to an American surrogate mother, with long-term partner, millionaire Ty Comfort.
When she was about six months into her pregnancy, the 41-year-old spoke about becoming pregnant for the first time, and about expecting two babies at the same time – one via a surrogate mother 5,000 miles away, and one being carried by herself.
Her second son, whom she is carrying, is expected to be born in about a month.
"I've waited 41 years for this, I can't wait!" she said in June, speaking to presenters Holly Willoughby and Philip Schofield on ITV's This Morning.
Caprice told of her joy at unexpectedly becoming pregnant after years of struggling with fertility issues and suffering a miscarriage, and having already chosen a surrogate mother.
"In March I found out I was pregnant," she said. "I remember taking a test and thought somebody upstairs is really taking the mickey, but I went to a store and bought every single brand of test and they were all positive. It was one of the best days of my life!"
Caprice went on to describe seeing her baby's heartbeat for the first time.
"I don't usually cry like this!" she said, overcome by emotion. "But I promise you, when you look at that screen and when you see that heartbeat after everything you've gone through, the feeling of happiness... I was so grateful."
Caprice and Ty had been trying to conceive and went through IVF treatment three times. At her doctor's advice, the couple explored their options and decided to look for a surrogate mother.