"We received the test results which confirm that the man in his 30s who died last Friday had H1N1 flu," the Akademiska hospital said in a statement.
The man had flu symptoms and died at his home near Uppsala, 80 kilometres (50 miles) north of Stockholm. He had not been travelling abroad.
Several people with the A(H1N1) virus are currently seriously ill in Sweden, where more than 900 people have been infected with the virus.
The other Nordic countries - Denmark, Finland and Norway - have so far not reported any deaths from the disease.
Sweden has already signed an agreement with a laboratory for a delivery of 18 million doses of vaccine for its 9.3 million population: the treatment requires two doses.
The first batches are expected early next month and will be on a voluntary basis, even if Sweden wants as many people as possible to take up the offer.
More than 2,180 people around the world have died from the virus since it emerged in April, according to the latest WHO figures, with an estimated 210,000 people infected in 177 countries.