I would have liked to know if one can transform an armchair covered with a fabric (bergère style) into a cane chair?
This is not desirable on this type of chair because its structure does not really lend itself to it. It is a very sharp job that requires the expertise of a professional upholsterer especially to get the correct tension of caning. If you really want to change your chair, give it to a specialist. Otherwise, be aware that there are models of armchair style shepherdess cannes available for sale (at Roche Bobois for example). If your chair has no sentimental value, it would be less expensive and especially less complicated to acquire a new one directly.
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