It was flying a group of 17 Swiss nationals and a three-member family from Austria on a sightseeing tour for the company JU-AIR.
The aircraft was a Junker JU52 HB-HOT plane, built in Germany in 1939 and now a collectors' item.
The 20 Minutes newspaper quoted a witness who was on the mountainside at the time of the crash.
The cause of the crash, which occurred hours after a family of four was killed when their small plane went down further west in the Alps, is under investigation.
Andreas Tobler, Grisons canton chief of police said there was "no longer any hope of finding anyone alive".
Daniel Knecht of the Swiss Transportation Safety Investigation Board said the plane appears to have hit the ground near-vertically and at high speed.
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JU-Air CEO Kurt Waldmeier told reporters that the plane had undergone maintenance inspection in July.
The company's flight operations were suspended, it said.
"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. It was used as a military transport plane as well as a bomber during World War II.
Ju-Air said one of its aircraft had been involved in the accident on Saturday.
Earlier on Saturday, a small plane carrying two parents and their two children crashed in the nearby canton of Nidwalden.