Britain has been harder hit by the virus than elsewhere in Europe. Earlier Sunday, Britain had reported another 61 cases of swine flu, bringing the country's total to 1,226 cases.
The Scottish government did not identify the woman who died, but British media said she was a 38-year-old woman who had given birth prematurely two weeks earlier. The baby was reportedly doing well.
"Tragic though today's death is, I would like to emphasize that the vast majority of those who have H1N1 are suffering from relatively mild symptoms," Scottish Health Secretary Nicola Sturgeon said. "I would reiterate that the risk to the general public remains low and we can all play our part in slowing the spread of the virus by following simple hygiene procedures."
Now that swine flu has, or global outbreak, health authorities expect to see more cases and deaths worldwide. The World Health Organization said last week that the virus has not become any more lethal, but is now unstoppable.
So far, swine flu appears to be a relatively mild virus, and most people who get it do not need treatment to get better. About half the people who have died from swine flu have had other health conditions including pregnancy, obesity, diabetes, or asthma.
"The patient had underlying health conditions," the government statement announcing the death on Sunday said, without saying what they were.
Scotland's government said the patient was one of 10 people being treated for the influenza at the hospital, identified by British media as the Royal Alexandra Hospital in Paisley.
Meanwhile, China reported 31 more cases of swine flu, bringing the total on the mainland Monday to 196.
The Health Ministry said on its Web site late Sunday that the cities of Beijing and Shanghai had reported more cases, as did provinces in central, eastern and southern China.
The world's most populous country has quarantined travelers -- on the slightest suspicion of contact with an infected person, and has increased surveillance of the virus with temperature checks on incoming passengers at airports.
There have been no reports of deaths.
Hong Kong, which tallies swine flu cases separately, said late Sunday it had confirmed eight more cases, bringing the Chinese territory's total to 92.
The World Health Organization said that as of last Friday, 74 countries had reported more than 29,000 cases of the virus, with 145 deaths worldwide. That tally did not include the death in Scotland.