First off, it's not the man, San Francisco musician Craig Zolezzi, who's been named in several press reports.
"Craig's not the dad!" the 6-months-pregnant star told the U.K.'s Independent Thursday. "He's my great friend and he's been so cool about it."
So who is it? The British-born Riches star – who is in the U.K. promoting her second album, Seastories – will only say that he's English and "sort of in the same business." (She also revealed that they have no plans to marry in the "near future.")
The former Good Will Hunting Star (who previously credited the Easter Bunny for her ever-growing bump) also said she's decided to keep the mystery going as long as possible.
"I want to shield the baby's dad as much as I can because it wasn't his choice to get roped into all this stuff," she told the paper – adding that her first pregnancy was "unplanned" but "very happily received."
"It'll come out in time," said Driver, who has a craving for a cheese sandwich every day at precisely 11:30 a.m. "There'll be some other bump people will be interested in and I'll have this baby and get on with my life."