|So while disabling the network adapter helps significantly, there is most likely a program, maybe a virus or trojan, that connects to it's home.|
So reenable the network card, open your favorit browser and browse to the web configuration interface of your router.
Hopefully you router has a statistics page, where it shows the current network activity.
If there is a huge data flow, while you do not run any program, that connects to the internet, there is an unsolicited guest on your computer.
In that case I suggest to download an antivirus livecd and check your computer, by booting from this cd.