So, I contacted the Sony Customer Relations for France.
They told me to get a quotation for a repair at their certified center and then send it to them.
The repair center said they could not fix the reader (problem comes for a main part which can't be replaced), but Sony agreed to take back my PRS-505... and gave me 212€ for it
! (I paid 300€ for it 3 years ago).
So I got myself the new PRS-T1 (available this week in France), and have still some money left for accessories (case, AC-adapter...)!
It goes without saying that I am more than happy with the way Sony handled this
Sony