Well, they don't really give you an IP, they give you a means to connect to your data.
So, if you were to ftp in to your site, you'd probably use their sub domain lets say
user123456.1and1useraccount.com (this is totally inaccurate, I am not sure what there scheme is)
you would enter your credentials and you'd have access to your data. If you hosted a domain you could just use that to connect. So you would enter mydomain.com and your credentials to access your data.
Hope this helps.
Well, of course they offer many other services, dedicated servers for example. So, depending on how much you want to pay you could probably get what you wanted.
Call them, they have an adequate customer service line with someone who will answer these questions more accurately.