Answer: this is a bug that was identified with CuteSITE Builder. We have isolated the cause, and thus discovered a workaround. The problem is bveing caused when you use the option to save a copy of your HTML files locally. Disable this option, and your files will not be deleted from your web site!
Here's how to do it:
1. Choose File > Publish website
2. Choose Options
3. Choose the HTML Directory tab
4. Uncheck the box to save to copy of exported HTML locally.
Note: we suggest that when publishing to the website, leave the above option unchecked. You can re-check it if you just want to preview the site without publishing. This creates a copy of all the HTML and image files on your computer.