Technically, it's not possible without leaving your site. If you want to post form information to another server outside of your site, you will need to submit the form to a page on that external site. In other words, the current page in the browser window is going to unload when the form is submitted. However, you can have the external site redirect back to a page on your site OR post the results on their page with a link back to your site. With some JavaScript code, you can even have the external site redirect back to your site with hidden form data (ie.. the results).
Give me some more specifics on what you are trying to do and I can probably provide some more detailed help both on ASP and JavaScript.
TW