According to some folks, there is a temporary workaround: see #4 and #5 here.
Proceeding with implementing the above temporary workaround, I’d like to first remove the changes from our prior attempt.
Please close terminal and open Synaptic and confirm that the r8168 is installed, as before, and that it is green (no broken package). You should also return your sources.list to its original form (remove the “universe” at the end of those lines that you added).
Now moving to the archlinux thread suggestion. Please follow the first few instructions here to get to the grub editor. On Aravisian’s instructions, do not make any change to quiet splash as he suggests. Instead, after quiet splash you will add noapic
Then close as he recommends (ctrl+x) and continue to boot.
The above will do a one-time boot change to see if the archlinux workaround actually fixes the problem. So, once you change this, do not shutdown/reboot, and put load on your network controller (for example, watch Youtube which has a high transfer rate) to see if it holds the internet connection. If it works we can do a permanent grub edit.