[TYPO3] Multiple Domains

[Lee M. Childress]

I am getting ready to work with multiple domains. I have a site, www.mysite.com. Within that tree I have a subfolder which I want a new domain, www.theirsite.com.
 
My question, if someone visits www.mysite.com, does www.theirsite.com (the page menu link) appear in www.mysite.com? Hope that isn't too confusing. Here's my goal: I am developing an information site for the county I live in. There are several businesses which I am hosting their sites within my site. I want menu links to their sites as if it was all one big site, but when someone goes directly to their URL, then only see their portion.
 
If there is one person out there I haven't confused, can I do what I want?