The problem with AMD and stability is not a CPU problem, it's a mobo problem, use a good mobo and you will not have any instability...
The mobo connects everything in the computer, so if it isn't well designed, then you will have all kind of problems, video, RAM, IDE controller, PCI and AGP bus.
This has nothing to do with neither the CPU or the chipset, or any other part for that matter. So stay away from the mobo's who have these probs (yeah, that's Abit).
Go for ASUS or MSI boards, those are usually the most stable ones...
I have recently built 16 servers running Athlons and T-birds, that's how much faith i have in the stability of an AMD system.
As for PIII, the only people pushing for them are the ones owning them, they just wanna defend their ego, so they don't look stupid for spending more money on less performance.