Kate Winslet and her husband Ned Rocknroll are currently nursing baby Bear at their home in West Sussex after welcoming him to the world at an NHS hospital a few weeks ago.
"They are all doing so well," a source allegedly told The Sun.
"They both absolutely adore him - and the name bear."
Winslet's spokesperson later confirmed the news that the fuzzy bundle of joy is indeed a baby Bear Rocknroll.
Bear joins sister Mia, 13, from Winslet's marriage to film producer Jim Threapleton, and brother Joe, 10, whose dad is director Sam Mendes.
Ned, 35, who works on his uncle Richard Branson's Virgin Galactic project, was previously called Abel Smith before changing his name to the offbeat musical title.
Last week, it was revealed that Kate Winslet has threatened Fathers4Justice with legal action after the group targeted her in a planned campaign that attacked her children's living arrangements.
The actress instructed her solicitors Schillings to issue the parents' rights activists with a legal letter in response to her inclusion in their yuletide advertisements, which sees her image appear alongside the caption: "Kate, every child deserves a father this Christmas."
Her lawyers claimed that the advert was "misleading and seriously defamatory," the Telegraph reported.
Her ex-husband, director Sam Mendes defended Winslet, telling the Daily Mirror: "It is inappropriate for this organisation to involve my family and I when they know nothing of our personal circumstances.
"Whilst I fully support fathers' rights, I can happily state this has never been a concern for me or my son."
The decision to use her image as part of the campaign stemmed from comments Winslet made about her children's living arrangements during an interview with Vogue in October.
"My kids don't go back and forth," she said. "None of this 50/50 time with the mums and dads - my children live with me, that is it."
Fathers4Justice said: "We are running this advert as part of a campaign to highlight the sensitive issue of four million children waking up on Christmas Day without their father.
"It is clear from what Kate says that she does not support 'shared parenting'."
A spokesman for Schillings confirmed that the firm had been instructed to act on behalf of Winslet.
Representatives for Winslet have declined to comment on the matter.