It left Locarno Air Base around 16h CET and was heading for the suburbs of Zurich when it crashed at an altitude of 2,540 metres.
Police spokeswoman Anita Senti told reporters that it was with "sad certainty" that "the 20 people aboard perished".
Police have given few details but have confirmed the crash, saying the aircraft came down on a mountainside about 2,540m (8,333 ft) above sea level.
Eleven men and nine women were killed, officials said.
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Police said they were not aware of any distress call from the aircraft before it crashed.
Deceased passengers include Swiss citizens and a three-person family from Austria, police say.
The company JU-AIR has three of the vintage Juncker propeller aircraft, which are known affectionately in German as "Auntie Ju" planes.
The vintage aircraft crashed into the Piz Segnas mountain in the Flims resort on Saturday.
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The airspace above the crash site was closed by the Federal Office for Civil Aviation and access to popular hiking trails in the surrounding area was blocked.
Collision with another aircraft and hitting an obstacle have been ruled out as a potential cause. Its pilots are ex-military and professional pilots, all of them volunteers.
"Yesterday was the worst day in the 36-year history of JU-Air", the airline's Chief Executive Kurt Waldmeier said at a news conference in nearby Flims on Sunday.
The plane that crashed had logged 10,187 hours of flying time, he said, adding that it underwent maintenance after every 35 hours of flight - most recently at the end of July.
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The company, which operates two other Ju-52s, suspended flights until further notice after the crash.
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