With the unusual first name the baby joins TV survivalist Bear Gryls, and Jamie Oliver's son Buddy Bear.
The baby was delivered on 7 December at an NHS hospital in Sussex. He is the first child for 35-year-old Ned, who is the nephew of the Kate's friend, Virgin mogul Richard Branson.
The 38-year-old actress already has two children – 12-year-old Mia, whose father is director Jim Threapleton and Joe, nine, from her marriage to filmmaker Sam Mendes.
Bear made his entrance a year after Kate and Ned married during a secret new York wedding.
They began dating in 2011 after meeting at Sir Richard's idyllic Caribbean hideaway, Necker Island. She was clearly attracted to his fun-loving spirit.
Formerly Ned Abel Smith, he changed his name by deed poll in 2008 to RocknRoll, "as a fun reminder to make sure every moment of every day is spent wisely and enjoying life".
Their union came a year after Ned, who works at Virgin Galatic – his uncle's space travel project – split from his wife, aristocrat Eliza Cowdray.
Despite being head over heels in love, Kate has said she's going to stick with her surname. "I was never going to change my name to Rocknroll," she said. "I've never changed my name to anything so I didn’t see a reason to start now."