Comment 18 for bug 914439

I'm facing the same problem after upgrading from oneiric to precise.
The following workaround works for me till this is fixed upstream:

There is a debian package in sid which provides the latest Realtek driver (8.029.00) including dkms configuration

http://packages.debian.org/de/sid/all/r8168-dkms

The nice thing about this is that it automatically re-compiles the r8168 module on kernel updates.