The actress gave birth to a little boy after a caesarean section on September 29 in her hometown Liverpool. Speaking through her rep, she said: "My baby is beautiful and has me awake every hour breast feeding."
Claire had previously said she would call the baby Gabriel but told a well-wisher on Twitter that she and the baby's father hadn't decided on a name yet.
The father is Claire's former fiancé Daniel Reilly, who is 14 years her junior. The couple became engaged in a romantic Christmas Eve proposal in 2012 and planned a wedding in the House of Lords.
Speaking about their relationship, the former Brookside actress told HELLO!: "I've had a great life and a fabulous career. It would take a very special man to make me stop and want to be with them. Daniel is that person. I was thrilled when he asked me to be his wife."
Sadly, the couple separated unexpectedly in 2014, but a month later Claire found out that she was pregnant.
Despite the split Daniel, a software tycoon, was delighted with the news and promised to take care of his son. The businessman posted this message on Twitter: "I'm going to be a Dad! Love you already son. Blessed."
TV favourite Claire will certainly need his help as she is a self-confessed workaholic who never sits still for long.
In her varied career the bubbly brunette has starred as Roxie Hart in the musical Chicago and been a presenter on TV shows Loose Women and 60 Minute Makeover.
Most recently she wrote and starred in comedy musical Sex and the Suburbs, about "men, dating and finding Mr Right".