Even though WoW has 512MB set as the minimum requirements, it will still have problems compared to 1GB. That's why 1GB is "recommended" by Blizzard.
If you do get a new computer, I would recommend something loaded with a Core 2 Duo CPU. Anyone will run games just fine, even if you don't tweak it.
If you want to play WoW, get something with 1GB and a decent PCI Express (PCI-E x16) video card (I wouldn't get anything less than a 7600GS or X1600 pro/XT). This game isn't too CPU intensive although I still recommend the Intel Core 2 Duo (C2D).
Now, whether you are going to build it yourself or buy pre-built is another question. The majority of people here will tell you to build it yourself, but that is mainly because it's cheaper and you know what your getting and I am included in that majority. However the question comes down to you. Doing it yourself isn't as nearly as hard as the average person out there thinks it is.