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First thing to try when a fitbit misbehaves is a restart using the following procedure:
https://help.fitbit.com/articles/en_US/Help_article/1186/
It does no harm and can sort out all sorts of issues. You might need to give it a few tries though as it's known to not always work.
Best AnswerI have the same problem. I have the Fitbit One. I haven't had any trouble with it for over 2 years. However, since it updated yesterday to 6.64, it won't stay on. When I put it on the charger, it shows that it is fully charged, but as soon as I take it off it dies. I followed the directions to do a restart, but same thing. It dies as soon as I take it off the charger.