Thanks for everyone's comments.

trader_4 says...

Anyway, I ordered what's supposed to be the correct
replacement valve, which is this one:

http://www.ebay.com/itm/381506821667

What do the instructions say? I see both a copper
compression type fitting nut and a plastic nut in the
bag. Plus the valve has two ends. I would think the
valve would accommodate both copper and plastic supply
lines.

The plastic nut in the bag is part of the extension needed
for the line to the ice maker. This is needed for some
older boxes like mine. With this new "universal"
replacment, the old line is no longer long enough.

I managed to get this done in only about three hours. Now
you're gonna ask why it took that long, but if you ever saw
me trying to do plumbing, you would understand.

First, I had everything backward. From the instructions,
which weren't for this valve anyway, it looked like the
standard compression fitting was for the line to the ice
maker, and the quick connect was for the supply line.

But after trying lots of different positions, and trying to
see how this would actually work, it became apparent that I
was wrong. Then, just glancing at the valve in the right
light, I saw a barely visible direction-of-flow arrow. So,
the compression fitting is actually at the supply line,
which is where I think it should be, and the quick connect
is for the line to the ice maker, where even if it leaks it
won't leak much.

Then all I had to do was figure out exactly where the valve
would be mounted, add the extension to the ice maker line,
connect everything, and hope it wouldn't leak. And of
course hope it would work.

Anyway, it did the first fill cycle successfully, and so far
nothing leaks. That's a good sign. But I'll leave the
fridge pulled out for a couple days so I can make sure it
isn't leaking. Then I may have to adjust the ON time for
the correct fill level.