Okay, I fixed it.
No doubt an HTML wizard would have simply inserted a page break command of some kind between the table of contents and the first chapter.
Me, I'm not an HTML wizard. NCX and ADE is all Greek to me. So I went back to the original InDesign manuscript file, and went to the page with the table of contents on it. I squeezed the text box containing the table of contents upwards just a little bit, leaving some room at the bottom of the page. I then inserted a single-line text box in this space. Inside that text box, I added an InDesign page break command.
Ran it through the Kindle exporter, and converted the Kindle to EPUB with Calibre. Bingo. The table of contents and Chapter One are now on two separate parts.
Thanks to JSWolf for the help and advice. It would have been more useful if I were more knowledgeable!
Last edited by tbrosz; 08-21-2012 at 12:14 AM.