It definitely supports the card, since I'm using it
Windows, linux, all OS's use drivers to access hardware, so you need a one of those. They're called modules in linux.
Now, depending on your distribution, you will get a set of precompiled ones for the most common hardware. You will have them in /lib/modules/<kernel version>/. Look for a file called something like rtl8019.o in one of the directories under there. If your dist didn't come with an RTL8019 driver then you need to get it elsewhere.
I don't quite remember if the driver comes with the linux kernel source, I think it does, so if you know how to compile the kernel or modules, you should check there. It's not that tough once you get the hang of it.
You can also check out the webpage of your NIC manufacturer and see whether they have a precompiled module, for an easier solution.
Lastly you can always check the web for that file, it's possible that someone else compiled a module like that that just might work for you.