Next month, Connelly plays a young mother in Dark Water, an edgy psychological thriller in which her big eyes widen over all manner of horror in a very strange apartment. But she didn't have time to be spooked by her London digs. She was rarely there.

"I think London is such a great city with kids," she says. "There's so much to do! You could do a whole kids' issue."

Most every morning, Paul was out the door, leaving Connelly and Kai with an extraordinary city to explore. If they were feeling lazy, which they rarely were, they only needed to cross a few streets and they were in the Princess Diana Memorial Playground. "Fantastic," she says. "It's in Hyde Park, sort of up by Kensington Palace. It has this massive pirate ship that's sort of plunked down in the middle of this sand pit. The whole thing is beautifully landscaped. There are these little wooden boats and climbing structures and plants and big musical instruments that the kids can play on. It's really kind of magical."

Kai loved that park, but not every day. He was six going on 16. "Quite precocious, which a lot of these city kids are," Connelly says. "He's playing Sebastian in their school version of Twelfth Night now. So they're studying Shakespeare. They'd already read some Shakespeare by the time we were there. He has a good sense of history, so he really enjoyed seeing the architecture of London and the little higgledy-piggledy streets and cars. And the cabs.

"Oh, yeah, Kai loves the cabs," she says. "All kids love the cabs. I love the cabs."

She'd let Kai hail one, and they'd fall into the back of a black London cab, its elegant interior big as a Buick, steered by the world's best cabdrivers. "They take their jobs really seriously and really have a thorough knowledge of the streets, which is really an accomplishment," she says.

"Where to, sir?" the cabbie would ask Kai and his mother.

Most often they'd direct the cab to a museum, because London has more museums than any city on earth. Connelly recites a litany: the Science Museum, "my favorite science museum, massive and really well-done"; the Victoria and Albert Museum, "which does programs for kids where they give you a digital camera and they let you walk around the museum taking pictures of sculptures. Then you come back and you find the sculpture that you want to transform yourself into, and they have this arts and crafts area where you make yourself look like the sculpture."

But frequently, Kai leaned more toward cannons than culture. "The Imperial War Museum," she says. "When you walk in, they've got old planes and memorabilia from the wars: posters and uniforms and swords, history of all the different wars. There's one area you go through and they have bunkers and trenches simulated with sounds piped in, air raid sirens in other parts."

They gazed upon superstars and royalty at the renowned house of wax,
Madame Tussauds, and its adjoining Planetarium, but left in a rush after one look at its Chamber of Horrors and its grim catalog of crime and punishment. "In the basement, they have a torture-chamber thing going on. We started to walk down the stairs and we both got entirely freaked out, and we left. Oh, I think they sort of simulate torture. I don't really know. It's a little horrific."

But both mother and son absolutely adored Hampton Court Palace, the almost-200-year-old, palatial country residence of the late Cardinal Wolsey, all-powerful archbishop of York. It's a universe unto itself with gardens and galleries, a palace, and a moat. "We ran through the maze in the garden," Connelly remembers. "It's a beautiful, beautiful place. You can't see it all in one day. I think we went there twice. I do remember Kai being very impressed with Hampton Court. When we were walking around there, he said to me, 'Mom, I think I know what I'm going to be when I grow up.'?"

"What?" Connelly asked.

"A man of leisure," he said.