Well, i'm not very sure on the PIII side, but if i was getting a PIII system now, i'd look and choose from the Asus CUSL2, P3V4X, CUV4X. Get a Slotkey.
Dunno about Abit, haven't seen much of the BE6II.
I dunno about the 1Ghz PIII. Maybe i'm miserly but i simply refuse to part with that kind of dough when i can get an Athlon system for less.
Abit MBs for Athlon systems seem to give more tweaking options. But if i guess you want stability and Asus's pretty good at that.
For the Athlon, as long as you've a good HSF installed properly like the Alpha or the FOP32/38, don't worry too much. These CPUs can stand higher temperatures than most. One trifling detail is the noise level which is why i added the FOP32 recommendation. I'm using it and it cools pretty well, not as well as the FOP38 but definitely less noisy. You'll understand when you hear the Delta fan in action.
Don't forget about the PSU. Most people tend to concentrate on the components but shave a few dollars off by buying a cheapo PSU which is pretty silly since your components get powered from there.
When getting Athlon,
You'll want a PSU that can provide at least 150Watts on the 3.3V and +5V rails. Also, make sure that the 5V standby current is at least 1 amp or more.
Although the +3.3V rail can sustain 14A while the +5V can sustain 25A, this rating is taken provided that only one of them is used, ie no load on all other voltage rails. If you use them together (just like in any PC setup), then the maximum current these rails can actually supply is lower than what they've stated here because some power will be drawn to the other rails as well (they're all interconnected).
So when you connect your PSU up to a PC, the maximum currents that your PSU can supply are not 14A and 25A anymore. So a much more accurate way of judging the load rating of the PSU on these two rails (and any others as well, but these two are most important because they power the CPU and motherboard) is to use the combined rating. What the combined rating means is that if both the +3.3V and +5V rails are being used, and as long as the current on any one of the rails does not exceed the stated current limit for that rail, then the total power (3.3*I1 + 5*I2) cannot exceed that combined rating regardless of current. The 5v or the 3.3 v are normally supplied to the CPU.
On the mobo is a voltage regulator which supply the one point what-ever volt to the CPU. These regulator can take the 5v or the 3.3 v to switch it to the required voltage.
The other reason that the 5v and 3.3v are interelated, the switching transformer have a secondary at 5v with a tap on the same winding for 3.3v . If you are familiar with transformer, you will understand this simple concept of the 5v seconday with a 3.3v tap.
Some good PSUs to get are Enermax 350W, Xenac 370W, Enlight HPS-300-101 300W.