Solution: an external IDE drive
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But here's where the cool part comes in. I googled for free backup software, and came across a forum posting that mentioned the good program DiscWizard that comes with Seagate hard drives. I thought, "hey I've got a Seagate hard drive, why don't I have that program". Next stop -- Seagate's downloads page for DiscWizard.
I looked at the software requirements, and the only requirement to use it was that you had to have a Seagate drive. Which fortunately, I do (a 250 GB IDE drive I bought to replace 2 80 GB drives that I never got around to actually installing). The program does check, via the hardware report, that the drive is actually made by Seagate; otherwise it won't work (I tested this with another non-Seagate drive that I had lying around -- yes, I have a lot of hard drives just sitting around. I'm a hardware geek).
<shameless plug>I should point out that the last 3 drives I've bought (the 250 GB, and 2 320 GBs) have been Seagate -- they all carry a 5 year guarantee, and they're usually the best reviewed ones on the web, in terms of performance and reliability.</shameless plug>
After the program installed and identified that I had a proper Seagate drive installed, I was able to make a complete backup image of my 80 GB laptop. But more than that, I could then make incremental backups (rather than overwriting the entire 80 GB image over and over again), and coolest of all, I could mount the created backup image as a drive on my laptop, and access the files inside it. The image mount can be either read-only, or even read/write.
That's pretty useful stuff for free software.