"The police have the sad certainty that the 20 people aboard perished", police spokeswoman Anita Senti told a news conference.
Initial investigation showed the plane did not lose any parts or break up before it crashed in the Swiss Alps.
"The JU-Air team is deeply saddened and is thinking of the passengers, the crew and families and friends of the victims", JU-Air said on its website on Sunday.
It crashed about 2,540m (8,333 ft) above sea level, on the western side of a 3,000m peak called Piz Segnas.
The parents and two children crashed roughly 30 minutes after their plane took off from an airfield in Kägiswil, headed towards France, Swiss police said.
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The area around the crash site, which is popular with hikers and skiers and includes a glacier, was closed to the public.
He and senior police official Andreas Tobler said the vintage plane lacked "black boxes", the crash-resistant cockpit voice and data recorders that more modern aircraft have.
Daniel Knech of the Swiss safety investigation service SESE said the crew did not have time to send out a distress signal.
There were 11 men and nine women aboard, including an Austrian couple and their son, she said.
JU-Air runs a small fleet of four Junker planes, all built in 1939 and operated by ex-military and professional pilots.
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He said officials have essentially ruled out a collision with another aircraft or hitting an obstacle such as a wire.
The company's flight operations were suspended, it said.
Based out of Duebendorf in canton Zurich, JU-AIR offers sightseeing and adventure flights around the Alps.
It said it was suspending flights until further notice.
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