Easy one.  Demux, Remux.  Use Avanti GUI front-end to FFmpeg.

Demux Video-only to a stand-alone file

1. Load the Original Source video into Source 1
2. Click on the toolbar icon in the top left corner, and select Load Template.
3. Load the file "UNI_DEMUX_SETUP.tpl"
4. Set the Destination filename to something appropriate (don't overwrite your source)
5. Click "Start process"

Demux Audio and Re-Encode to MP3

1. Load the Original Source video into Source 1
2. Disable the Video Settings checkbox
3. Enable Audio Settings, and configure as you like
4. Set the Destination filename appropriately
5. Click "Start Process"

Remux Video and Audio to a new File

For whatever reason, I can't get Avanti to remux well.  The video is afu.  Not sure if this is a problem with ffmpeg, or avanti, or just my settings.  I might try to figure this out one day, but what's the point?  VirtualDub is here!

1. In VirtualDub, load the video-only AVI file by going to File> Open Video File
2. Unless you want to recompress the video (you're welcome to, but that's not what we're shooting for in this guide), turn of video compression by going to Video > Direct Stream Copy
3. Now load the Audio track by going to Audio > Audio from other file...
4. Again, make sure you're not recompressing the Audio.  Go to Audio > Direct Stream Copy
5. Now go to File > Save as AVI, and enter a new filename

The process is quite fast.  And with that, I realise that I'm an idiot, and that all of this could have been done directly in VirtualDub.  HOWEVER, VirtualDub on most people's systems will only use the MP3 encoder library built into windows, which is only capable of shit bitrates and frequencies.  If anybody wants to email me the way around this, or a way to get virtualdub to use LAME, I'm all ears...