I found lots of instructions on how to do it from a Windows PC, but I have already started to use the Ubuntu Desktop version 10.
I found detailed descriptions and lots of command lines in the Ubuntu documentation - and got the feeling this could take some time.
Then I saw the "Startup Disk Creator" under System > Administration and that was it! I had already downloaded the disk-image. I had my USB ready. I clicked on "Erase Disk", then on the "Make Startup Disk", and waited ...
Once the USB was ready - all I had to do was to get the BIOS screen up when starting my dead little notebook. (Had to google this: F2 and Esc on my Eee PC). Made sure it would boot from the USB - and it did!
The installation of Ubuntu is so easy! And my Eee PC is working again ;-)