On Windows, using a program similar to Audacity, I could do this, but if I start an internet audio stream, then open Audacity, It gives me a message that since the sound card is being used, I won't be able to play or record audio.

Reading around, it seems like the solution is the Jack audio program, but I can't seem to get that working on either my 32 or 64 bit systems. I keep running up against suggestions that I'll need to recompile my kernel. I haven't found a really approachable (read simple) howto for getting Jack working on a Debian system. I think I'm capable of rebuilding kernels and fighting thru the process of getting it to work, but I'd rather have an easy way. (Who wouldn't?)

I'm considering doing a fresh install using 64 Studio which apparently has the kernel modifcations required for that sort of thing already in place. Since it's a Pure Debian OS, it seems like I'd be able to take advantage of their advanced audio functionality, and just add back any apps that they don't include from standard Debian repos.

Does that sound reasonable? Have I missed some simple solution to the title question?

Well maybe I am not understanding what you want to do but wouldn't streamripper http://streamripper.sourceforge.net/ do what you need.

Streamripper is fine for an icecast type stream, but my problem is broader than that. I want to be able to record whatever the soundcard happens to be playing. There is an Audacity instruction that says I should be able to select "wave" or "mixer" as my input and that it should work. (That's what I did on Windows) But I don't get that option on my mixer tool with Audacity.

Seems to me that I've read about Alsa being limited to one input at a time, but I don't recall seeing a solution.