Switch on the computer so that it *doesn't* connect to the Livebox, then run IPCONFIG /ALL in a Command Prompt window:
Go to "Start > Run" type...
cmd.exe /k ipconfig /all
...and press Enter (note the spaces). This will show details of all network adapters in the PC. Right click in the Command Prompt window, "Select All" and press Enter to copy the details to the clipboard. Paste the details into a forum reply here once you have regained a connection. If you need to reboot the PC to get a connection, paste the details first into a Notepad file so they aren't lost. This might just give a clue as to what is happening.