The submarine, the UC3 Nautilus, 26 feet long, was found 22 feet below sea level and was brought ashore shortly after it sank.
Using divers and sonar, the authorities were searching for Ms. Wall’s body along the submarine’s route, north and south of Copenhagen, the police statement said.
Mr. Madsen’s new account was only disclosed on Monday, with the approval of both the prosecution and the defense, although he had given it in court, behind closed doors, on Aug. 12, a day after his arrest. He is due again in court next month.
Mr. Madsen’s lawyer, Betina Hald Engmark, told the Danish television network TV2 that her client was cooperating with police investigators and that he maintained that he was not guilty.
The details were not immediately made public, officials said, to protect the police investigation and out of concern for Ms. Wall’s family. Her relatives have said they believed Ms. Wall, 30, had traveled to Denmark on a reporting assignment.
Friends of Ms. Wall have told local news media that she was about to move to China with her boyfriend. According to her friend and fellow journalist Victoria Greve, writing in the Swedish daily Expressen, Ms. Wall had signed a lease for a small studio apartment in Beijing.
A freelance journalist and a graduate of Columbia University in New York, she had written for a wide range of international publications, including The New York Times.
Ms. Wall had reported on Ugandan ghost stories and on Cuban internet pirates who distributed episodes of “Keeping Up With the Kardashians” without authorization. With Ms. Greve, she reported for Swedish Radio on affluent American women who support President Trump.
Mr. Madsen, 46, is known in Denmark as “Rocket Madsen,” an uncompromising builder of submarines and space rockets who was hoping to become the world’s first amateur space traveler riding in a homemade rocket.
For years he was able to build a community that offered helping hands and raised funds for his projects. But his temper caused conflicts with many of them, Thomas Djursing, a biographer, told BT, a Danish newspaper.
“He argues with every Tom, Dick and Harry,” Mr. Djursing said. “I’ve argued with him as well. But that’s what it’s like with people driven by deep passion.”
In her Expressen article about Ms. Wall, her colleague, Ms. Greve, reflected on how improbable it was that a short day trip to Denmark would end up being the last reporting trip of her friend’s career.
“There’s a dark irony in Kim, who traveled to North Korea and reported from Haiti, should disappear in Denmark,” she wrote. “Perhaps it speaks to the vulnerability of female freelance journalists. To work alone and do everything. Kim can photograph and shoot film as a complement to the texts.”
For some Danes, the mystery had the air of the Scandinavian crime thrillers for which the region is known. “This story is endlessly fascinating and as in any good crime novel we find the truth piece by piece,” said Lone Theils, the author of “Fatal Crossing,” a novel about two young girls who go missing on a ferry.
“There’s still a lot of mystery and lots of speculation. Everybody here has their own theories on what happened. I haven’t been anywhere at dinner or coffee with more than two people where this story didn’t come up. People share what they’ve heard and what they think.”
“Denmark is such a small country and everybody feels close. A colleague of mine’s brother has a friend who knows him,” she said, referring to Mr. Madsen, “and another colleague knows Kim Wall’s parents.”
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