Linux kernels 3.2 and later have support for this adapter built-in (with config USB_NET_AX8817X).

For others, compile/install steps on Ubuntu (required each time the kernel is updated).

Details: http://plugable.com/2010/10/18/howto-asix-88178-usb-ethernet-adapter-on-ubuntu-10-10-linux/

Steps:

mkdir asix
cd asix
copy source on CD to this directory
sudo apt-get install module-assistant
sudo module-assistant prepare
sudo modprobe -r asix
make
sudo make install
sudo modprobe asix