I've done this:
See the "Day 7" hack for my job. I did it the hard way - with hand tools (hacksaw, jeweller's saw, file) and far too many hours of slicing. What I wouldn't have given for 10 minutes with a milling machine.
The CPU runs hot. I figured if I could keep the CPU sensor to under 60C, I'd remain within the operating range of the hard drive. But it's marginal.
My unit also has (Day 6 pics) an insanely tiny 5V fan mounted near the heatsink. Your breath probably moves air than it does. But it drops temperatures a couple of degrees, and only cost a couple of bucks at the surplus store. And unlike a "real CPU fan", this thing is *silent*. The laptop drive is louder than this fan.
Anyways, I've run my IO for several hours with hard drive mounted in the heatsink at 200 MHz with the original WinChip, and with a K6-III-333 underclocked to 200 MHz at 2.1V. Works fine, and heat remains "warm", but manageable. With software-based utilities like CPUCool, heat drops from 55C to 45C, which is on the warm side, but safe for the hard drive.
If performance isn't a huge issue, underclock the K6-III a little more. A K6-III at 166 MHz runs several degrees cooler than the original WinChip at 200, and is still faster. (There's a long post of mine in the "very technical" section from a weekend of torture-testing an IO with various CPU configs.)
The rest of the info in the thread is also right: turning the heatsink so the fins are aligned vertically would improve airflow; my "mini-fan" is the other solution. Cutting out such a large chunk of heatsink didn't increase temperatures all that much. CPUCool, Rain, etc. are your friends. The slower (and cooler) you run the CPU while still being able to get the job done, the better.
Umm. One thing to add. Find someone with a milling machine. It's a *lot* of work with hand tools. Don't even think of grinding out that much stuff with your Dremel. (A Dremel is great for final sanding, though.)
If you do it yourself, do it *far* away from any electronic components, and wash up thoroughly. Water won't hurt the heatsink pad (assuming the pad stays on the heatsink and not on the CPU when you remove it). Metal filings will kill your IO. (They're also none too healthy for your eyes - use safety goggles when cutting and wash your hands at least as well as you wash the heatsink!)