Question: I have an account at an ASP hosting company and the URI is: me.example-host-company.com
I'd like to my own domain name so I create a CNAME record for a subdomain which has its destination set to the above URI. 

So 

```
CNAME:  subdomain.mydomain.com ---> me.example-host-company.com

```

The problem is only the main welcome IIS page of example-host-company.com is shown when I point my browser to subdomain.mydomain.com, and not the application that I deploy to me.example-host-company.com

For this host plan which only costs a few $/month, I don't get a dedicated IP address. So with what I want to accomplish, is that possible and how to do it?

Answer: 
Based on your description, it sounds like you need to configure a host header name for your web site in IIS. Since you didn't mention which version of IIS you're running I'll give you a link for IIS 6 and IIS 7:

<http://www.microsoft.com/technet/prodtechnol/WindowsServer2003/Library/IIS/883a9544-3f70-4d46-a6df-bbadbd1fe7de.mspx?mfr=true>

<http://technet.microsoft.com/en-us/library/cc753195(WS.10).aspx>
The browser will still use the original URL in its request, which is unknown to the webhosting server.
You would need to add that subdomain hostname to the me.etc website as an alias - how to do this through a hoster depends on them.