Eyewitnesses said the park's Jetline roller coaster had partly derailed during a ride, sending people crashing to the ground.
"Today is a day of mourning at Grona Lund, we've had a very serious accident in the roller coaster Jetline, where one person has died and nine people have been injured," Jan Eriksson, the park's chief executive, said in a press conference.
Ambulances, fire trucks and a helicopter were seen arriving shortly after the accident, and police launched an investigation.
Police said the nine injured people were being treated in hospital, and that three had severe injuries.
"Something like this should not happen at Grona Lund, and yet it happened", Eriksson said, adding that the 140-year-old park would be closed for at least a week to aid the police investigation.
The park spokesperson said 14 people were on the roller coaster when the front part partially derailed. It then stopped in the middle of the track with one carriage leaning out.
Jenny Lagerstedt, a journalist visiting the park with her family, told Swedish broadcaster SVT she was nearby and heard a metallic noise and noticed the track structure was shaking at the time of the accident.
"My husband saw a roller coaster car with people in it falling to the ground," Lagerstedt said.
"My children were scared," she added.
Grona Lund is a popular attraction on the waterfront on one of Stockholm's many islands, surrounded by several museums.
The steel-tracked Jetline roller coaster reaches a speed of up to 90 kph (56 mph) and a height of 30 metres (98 feet), transporting more than one million visitors each year, the amusement park says on its website.
Sweden's Culture Minister Parisa Liljestrand said news of the accident was incomprehensible.
"My thoughts are with those that were affected as well as their families and loved ones," Liljestrand said in a statement to the TT news agency.
