If you would like to start a new site, with a separate domain from your main domain name, you will want to set up that domain as an Addon domain in your cPanel account. If this is a new domain, not previously registered, you will need to register that domain before creating it as an addon domain. Once you have created the new domain as an addon, you can then install WordPress. Be sure to select your addon domain in the Install on domain drop down menu in Fantastico.
I'm not clear on one part of your question, you said "with a different domain name, but keep my blog address." Can you please clarify what it is you want to do?
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