I finally got the solution, here's what I did :
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based on the comments of this person, I uninstalled MikTeX and the complete TeX distributions on my computer. Since I use Ubuntu, I used purge to remove MikTeX, TeXLive, tex-commons, etc.
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then, I installed (or re-installed) TinyTeX (details here)
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I installed manually the packages I needed with tlmgr_install(). Normally, it can be done automatically but that's what I did.
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finally, I replaced biber by bibtex in \usepackage[backend = biber]{biblatex} (this person explains why it is important)
This last step made it work, but I don't know if the previous ones are important or not. Since he succeeded in compiling the bibliography too, I suppose it is necessary but there's no way for me to prove it.
Anyway, now, when I compile the Sweave document, the bibliography is directly created. Thanks a lot to this person for his/her ideas.
Edit : I re-installed MikTeX after all of this and it works too.