No, you cannot use DropBox to serve the Django website, as this requires running Python code when DropBox is only for saving and maintaining files.
You can, however, find free hosting for your Django needs, including:
Heroku , originally designed for Ruby hosting, but now allows you to do Python hosting. I did not find it particularly easy to use with Django.
ep.io , Python-specific hosting, but invite only for the moment - you can get it by sending an email explaining your project and your needs. Offers you free SSL on * .ep.io, which is pretty nice and they have great service.
Dotcloud , not platform specific, but my personal favorite. It's pretty easy to get started, and the CLI allows you to do a lot (but the disadvantage is that you need it too). If you are not afraid of the CLI, that's great.
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