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The defeat leaves Darmstadt, coached by ex-Germany midfielder Frings, nine points from safety with two games left after Juan Bernat scored Bayern's winning goal in Munich.
The Lillies' two-year stay in Germany's top flight will end after next Saturday's home game against Hertha Berlin and their away match at Borussia Moenchengladbach on May 20.
The team which finishes 17th, currently Ingolstadt, will join Darmstadt in the second division next season.
The team which finishes 16th, currently Hamburg, faces a two-legged play-off against the third-placed team in the second division on May 25 and 29.
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