Tested everything (from the posts above) in IE6 and IE7 and verified what you wrote apart from when I replaced:
<input type="submit" name="submit" value="Then the next time, click this">
<INPUT TYPE="image" SRC="images/signup.png" HEIGHT="25" WIDTH="63" BORDER="0" ALT="Submit Form">
An image in place of a submit button has no value. The very code you show has no value to it and no it won't be sent.
Not true. In reality (in both IE6 and IE7) when I pressed enter it actually came back with:
Your data was processed!
Which indicates that it is sending some kind of value and the bug doesn't occur if you use an image!
Would love to hear thoughts on this as this is the solution I have to use..