I found my own answer by snooping around on Google...
1. Create a page called "nonewpage.php" containing the following code:
<?phpheader(”HTTP/1.1 204 No Response”);?>
2. Upload the new page to your server.
3. Set the redirect link to the URL of the nonewpage.php file.
The header code tells the browser that there's nothing to see at the redirect URL, so the browser stays on the same page.
More details here:
http://biztoolsbrief.com/2007/06/20/tiny-scripts-for-stealth-form-submissions/