Johnson is engaged to Sam Taylor-Wood, a 44-year-old artist who directed his film "Nowhere Boy" in 2009. They also are the parents of an 18-month-old daughter named Wylda Rae.
E! News reported Wednesday baby Romy Hero was born Jan. 18 in London.
Taylor-Wood also has two daughters with her ex-husband, art dealer Jay Jopling.
A representative for Johnson and Taylor-Wood told E! News "both mother and daughter are doing well."
Johnson's film credits include "The Illusionist," "Kick-Ass" and "Albert Nobbs."