I got some quite good results with a celeron 450 based PC
It depends what you want to do really, if you want 100s of tracks, loads of realtime plugin effects and software instruments you'll need something more powerful, but if you're happy with 4-6 tracks, and the minimum number of plugins you may even be ok (but 5gb won't go very far..)
As far as power goes, I think any reasonably modern spec will be ok for all but the most demanding users. My last PC was an Athlon 1700XP with 256mb RAM, and I never ran out of horsepower when recording.
This is very debateable and liable to get me flamed but I'll say it anyway.. In my experience
Intel based systems have been a lot more stable, and easy to get going than AMD based. I've had great results with Athlon based systems, but always after spending many hours tweaking and troubleshooting. The intel based systems generally have been a lot more trouble free.
I'd recommend buying a PC from a specialist music pc retailer, as they know what components work well together, and will preinstall the OS with optimisations for recording. Some places I've seen recommended are:
Also, if you're going to be using the PC as a general office/internet/games machine as well as for audio, I'd partition the drive and have a seperate OS install for music only. I've been doing this for a while, and found it makes a big difference, in performance and stability. My recording partition has all un-necessary hardware disabled, and contains nothing but the OS, Logic Audio Platinum and one or two other audio apps.
Hope that helps!