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You want two separate domain names to refer to virtually the same HTML files and images except on 'domainA' there is a slight change in the logo on the page and a few links are to be different.
You don't want to copy all the files into a new directory but want some sharing of the sites.
There are various ways to achieve this. However, to give a good answer we'd need to know:
a) which web server you are using (Apache, IIS etc.)
b) whether you have access to the configuration files of the server
c) what (if any) server side scripting languages you have available (PHP, Perl, ASP etc.)
d) whether you have access to DNS (domain name service) for your domain and can create sub-domains for it.