In fact, “The Killing” has been remade for the United States, and that one will have its AMC premiere on April 3. The American version hews very closely to the original, with the same three-strand plot and with characters modeled on the Danish ones, said Joel Stillerman, AMC’s senior vice president for original programming, production and digital content.

“We tried to embrace a lot of what we thought made it incredible, including the Nordic sensibility, the stoicism of Sarah Lund and the lack of that overtly frenetic behavior that you’re constantly seeing on American crime and police shows,” Mr. Stillerman said. “Instead of having a chase scene with a standard bunch of cop cars with their lights flashing, we have things that you’d be much more likely to see in a horror movie — a scary walk down a dark hallway with the right piece of music.”

The series was a phenomenal hit in Denmark, where the final episode had a roughly 75 percent market share. (There was a second similarly popular season; a third is being discussed.) The distributors have sold it to broadcasters in, among other places, Australia, Austria, Switzerland, the Netherlands and Belgium — both the French and the Flemish-speaking parts.

In Britain the show has been averaging about 500,000 viewers an episode, a huge number for BBC4, a station with a generally tiny audience, on a Saturday night. (Its ratings are higher than those of the same channel’s showings of “Mad Men.”) The series has made an international star out of Sofie Grabol, one of Denmark’s most celebrated actresses, who plays Sarah Lund. It has been a stretch, Ms. Grabol said.

“I had always played very emotionally expressive women,” she said when interviewed by telephone backstage at Denmark’s national theater, where she was rehearsing “Fanny and Alexander.” But this time, invited by the producers and writer to help conceive the character, she told them that she wanted “someone who was not communicative at all, who was very isolated, but at peace with it.”