The funny man, 41, is said to have announced the happy news at a gig at the Nottingham Playhouse, telling the audience he 'needed to get down the f***ing M21 as my girlfriend has just had a baby'.
Fan Daniel Crann, 29, exclusively told the Mirror: 'He didn't confirm if it was a girl or a boy.'
Representatives for Russell and Laura have yet to comment on the birth, but that hasn't stopped fans tweeting their congratulations, with some even alluding to the sex of the child.
'@rustyrockets I hear you have a baby girl. Many congratulations to you and Laura!!x' one tweeted, whilst a second added: 'aw apparently @rustyrockets has had his baby, how cute! congratulations'.
Russell confirmed his pregnancy news with Laura, the younger sister of TV presenter Kirsty Gallacher, back in July, sharing a snap of himself reading 'The Expectant Dad's Handbook' to his Instagram account and writing: 'Right then. My Mum bought me this. Time to get ready! #trewlit #bookstagram #parenting'
He's since been vocal about his excitement at becoming a first time dad, telling Jonathan Ross in a recent interview that he's been busy preparing for the new arrival.
'I read a different book every week and I read one on preparation for fatherhood to mentally prepare me for it,' Russell explained.
'I've painted the room, I've done the things that you're supposed to do, I've accepted the idea that that person will be more important than me.'
When asked if the pair had come up with any potential baby names, he joked that he was considering 'Jesus Three'.
'We don't know the gender,' he added. 'I may not even ever impose a gender upon it, let the child grow up and be the whatever the hell it is, never tell it there is such a concept.'