How to use a subdirectory as a root for domain

I'm very new to setting up web.config files so I'm hoping this is a simple fix.

I have two domains for an account:
www.primarydomain.com - points to root
www.secondarydomain.com

I'd like to have secondarydomain.com have its files in a subdirectory off root:
/secondarydomain

I read this article:
http://blog.discountasp.net/root-redirection-to-subdirectory/

I updated the web.config file as such:
******************
<rewrite>
<rules>
<rule name="secondarydomain" stopProcessing="true">
<match url="^$" />
<conditions>
<add input="{HTTP_HOST}" pattern="^(www.)?secondarydomain.com" />
<add input="{PATH_INFO}" pattern="^/secondarydomain/" negate="true" />
</conditions>
<action type="Rewrite" url="/secondarydomain" />
</rule>
</rules>
</rewrite>
******************

It works but it displays the subdirectory name when you hit the url:
www.secondarydomain.com/secondarydomain

I don't want to show the subdirectory folder name (/secondarydomain).

How do I fix my web.config to use /secondarydomain as if it were the root directory for www.secondarydomain.com without showing that secondary folder name in the URL?

 

This caused a 404 error. Here is the detail:
Detailed Error Information:
Module IIS Web Core
Notification MapRequestHandler
Handler StaticFile
Error Code 0x80070002
Requested URL http://www.secondarydomain.com:80/subfolder_name/subfolder_name/
Physical Path E:\web\primarydomain\htdocs\subfolder_name\subfolder_name\
Logon Method Anonymous
Logon User Anonymous

I was able to update the code to stop the 404 by changing the match URL value to it's previous value:

Code:

<rewrite>
  <rules>
    <rule name="Rewrite to secondarydomain" stopProcessing="true">
<match url="^$" />
<conditions>
<add input="{HTTP_HOST}" pattern="^www.secondarydomain.com$" />
</conditions>
<action type="Rewrite" url="subfolder_name/{R:1}" />
</rule>
</rules>
</rewrite>

But it doesn't mask the /subfolder_name/ in the url (and also shows it with two forward slashes). Here's what I get when I hit the www.secondarydomain.com:
http://www.secondarydomain.com//subfolder_name/

 

Ugh...I just removed the rewrite rule and noticed the site still re-routes to that folder. Thus, somewhere there is a redirect already happening (which would then explain the double folder in the previous 404 error). I'm hunting down where that configuration is at (although I thought the only place could be in the web.config root folder file).

 

Just figured out it was a caching issue on my end.
Code you provided worked like a charm.
Thank you!