Here is what I wrote a couple hours ago:
OK, I'm ashamed.
I did what pomon said: it didn't work. So I decided to check with another Ethernet cable, and guess what ? It worked ! Meaning the issue wasn't my built-in Ethernet card, but the cable. The built-in Ethernet card is able to deliver 1000 Mb, and it does it very well.
I'm very ashamed, I thank everybody very much, and I apologize for all the inconvenience.
Solved !