NO. Your visitors will only see your custom site address (i.e. www.yourdomain.com).
Keep in mind, you will always have an IconoSites site address (the one you signed up with) - this is mandatory. However, nobody will ever see it unless you provide it to them. This includes the search engines.
Once you customize your site address (i.e. www.yourdomain.com), this is the only address you should provide.