Amsterdam city council says the diseased tree behind the building in which the Jewish Frank family took refuge has been attacked by a fungus.
But after protests by environmental groups and the Anne Frank Museum, a cutting of the tree will be replanted.
The chestnut tree is listed as a monument and is at least 150 years old.
The 27-tonne tree has been attacked by the horse chestnut leaf miner moth and by a fungus. It could be dangerous if it falls, so Amsterdam City Council says it has to be cut down.
The fungus rots the wood and has caused concerns about public health.
But after a lobby to save the Anne Frank tree, grafts were taken and a sapling will soon replace the original tree that she saw in the courtyard from her hiding place in the attic.
The new tree will be a big attraction for the Anne Frank Museum, where the Frank family's tiny flat is preserved.