David Lagercrantz will write a new story about the characters of Lisbeth Slander and Mikael Blomqvist, which is due for publication in August 2015.
Stieg Larsson died in 2004 aged 50 and had started work on a fourth novel.
The original trilogy sold more than 75 million copies worldwide and has been turned into both Swedish and Hollywood movies.
Daniel Craig and Rooney Mara played the lead roles in the US film version of the story, which was directed by David Fincher.
Eva Gedin from publishing house Norstedts said: “We have now decided to let somebody take over and tell what happened next.”
David Lagercrantz, who co-authored a book about Swedish footballer Zlatan Ibrahimovic, said he has “already started writing it”, adding: “It’s terribly fun. It’s a fantastic world to step into.”
The new book will follow on from the tale of murder, corruption and family secrets told in the original trilogy and feature its central players.
Some plot lines from the existing novels will be picked up, while David Lagercrantz is likely to add his own touches to the book.
Stieg Larsson’s partner Eva Gabrielsson has said that the publisher did not have access to Larsson’s partially-completed manuscript for a fourth novel, so would be starting afresh.
She told Swedish newspaper Aftonbladet that she thinks it’s “distasteful to try to make more money” from the books.
Eva Gabrielsson has been engaged in a legal dispute with his family over his legacy.
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