"The state of this monumental chestnut is a real danger for its surroundings," including the "secret annex" atop the canal-side warehouse where the Frank family hid, the city said in a statement.
"Its rapid decay makes it necessary to take action now."
The 150-year-old tree suffers from fungus and moths that have caused more than half its trunk to rot. The tree is familiar to millions of readers of "The Diary of Anne Frank."
The Jewish teenager made several references to the tree in the diary that she kept during the 25 months she remained indoors until the family was arrested in August 1944.
Anne Frank died of typhus in the Bergen-Belsen concentration camp in March 1945.
"Nearly every morning I go to the attic to blow the stuffy air out of my lungs," she wrote on Feb. 23, 1944. "From my favorite spot on the floor I look up at the blue sky and the bare chestnut tree, on whose branches little raindrops shine, appearing like silver, and at the seagulls and other birds as they glide on the wind. ...
"As long as this exists, ... and I may live to see it, this sunshine, the cloudless skies while this lasts I cannot be unhappy."
In March, the city council granted a license to have the tree cut down. That prompted protests by the Netherlands' Tree Institute and others who argued the tree had such an important historical value that extraordinary measures should be taken to preserve it.
It was granted a reprieve in October while the Utrecht-based Tree Institute investigated ways to support and heal the tree, but the City of Amsterdam said Tuesday that those ideas were unworkable.
"From the latest assessment, it appears that only 28 percent of the trunk is still healthy," the statement said. "The risk of the trunk breaking in which case the 27-ton tree will fall over is now unacceptably high."
"Given these results, which preclude the tree's being cured, preventative cutting of the tree is the only remaining realistic option."
The Anne Frank Museum, where the tiny apartment has been preserved, said grafts have already been taken and a sapling from the original chestnut will eventually replace it.
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