Realtek has released a new revision of the driver in january.
The release notes could be of interest (but I'm not sure it is related to our issue): 

release date: 2008/01/09
driver version: 8.005.00
1. Fix hibernate and cannot wakeup issue.
   The root cause: In rtl8168_init_board, tp->dev is never assigned. Therefor, it is a NULL pointer and the system crashes when this pointer is accessed.
   Solution: Assign a vaild memory address to tp->dev.
2. Modify GPHY parameters of RTL8111C
3. Modify GPHY configurations of RTL8111C
4. Create a spin lock to protect GPHY configuration.
5. Implement a kernel timer to rescue the NIC after the PCI reset is triggered.


Unfortunately the modules does not compile on my distro (Ubuntu Hardy Heron 2.6.24-11-generic) :

py@pc-fixe2:~/r8168-8.005.00$ make modules
make -C src/ modules
make[1]: entrant dans le répertoire « /home/py/r8168-8.005.00/src »
make -C /lib/modules/2.6.24-11-generic/build SUBDIRS=/home/py/r8168-8.005.00/src modules
make[2]: entrant dans le répertoire « /usr/src/linux-headers-2.6.24-11-generic »
  CC [M]  /home/py/r8168-8.005.00/src/r8168_n.o
/home/py/r8168-8.005.00/src/r8168_n.c: Dans la fonction «rtl8168_init_board» :
/home/py/r8168-8.005.00/src/r8168_n.c:2270: erreur: déclaration implicite de la fonction « «SET_MODULE_OWNER» »
/home/py/r8168-8.005.00/src/r8168_n.c: Dans la fonction «rtl8168_init_one» :
/home/py/r8168-8.005.00/src/r8168_n.c:2570: erreur: «struct net_device» has no member named «poll»
/home/py/r8168-8.005.00/src/r8168_n.c:2571: erreur: «struct net_device» has no member named «weight»
/home/py/r8168-8.005.00/src/r8168_n.c: Dans la fonction «rtl8168_rx_interrupt» :
/home/py/r8168-8.005.00/src/r8168_n.c:3790: erreur: «struct net_device» has no member named «quota»
/home/py/r8168-8.005.00/src/r8168_n.c:3790: attention : type defaults to «int» in declaration of «_y»
/home/py/r8168-8.005.00/src/r8168_n.c:3790: erreur: «struct net_device» has no member named «quota»
/home/py/r8168-8.005.00/src/r8168_n.c:3790: attention : il manque un transtypage pour comparer des types distincts de pointeur
/home/py/r8168-8.005.00/src/r8168_n.c: Dans la fonction «rtl8168_interrupt» :
/home/py/r8168-8.005.00/src/r8168_n.c:3986: erreur: trop peu d'arguments pour la fonction «netif_rx_schedule_prep»
/home/py/r8168-8.005.00/src/r8168_n.c:3987: erreur: trop peu d'arguments pour la fonction «__netif_rx_schedule»
/home/py/r8168-8.005.00/src/r8168_n.c: Dans la fonction «rtl8168_poll» :
/home/py/r8168-8.005.00/src/r8168_n.c:4035: erreur: «struct net_device» has no member named «quota»
/home/py/r8168-8.005.00/src/r8168_n.c:4035: attention : type defaults to «int» in declaration of «_y»
/home/py/r8168-8.005.00/src/r8168_n.c:4035: erreur: «struct net_device» has no member named «quota»
/home/py/r8168-8.005.00/src/r8168_n.c:4043: erreur: «struct net_device» has no member named «quota»
/home/py/r8168-8.005.00/src/r8168_n.c:4046: erreur: trop peu d'arguments pour la fonction «netif_rx_complete»
make[3]: *** [/home/py/r8168-8.005.00/src/r8168_n.o] Erreur 1
make[2]: *** [_module_/home/py/r8168-8.005.00/src] Erreur 2
make[2]: quittant le répertoire « /usr/src/linux-headers-2.6.24-11-generic »
make[1]: *** [modules] Erreur 2
make[1]: quittant le répertoire « /home/py/r8168-8.005.00/src »
make: *** [modules] Erreur 2
py@pc-fixe2:~/r8168-8.005.00$ 

I guess I have to set up an environment with an older kernel in order to compile and test this module. But if someone can advise me how to get this module compile in my current env ... 

PY