There is no technical solution for a social problem of trying to build a website that looks like yours, because no matter how you make the website look, every color, every font, every text and every picture that can be downloaded and displayed by your visitors can also be downloaded and copied by the imposter. There may be other than technical solutions to this problem and you should get some legal advice if you are serious about it.
That having been said, I can suggest few things that would make it harder to more expensive to look exactly as your site:
At Go Daddy you can purchase an SSL certicicate and you can set up DNSSEC extensions for your domain. It will make your site look clearly more legitimate than the fake site.
You can use some commercial fonts on your website and contact the foundry that you bought them from if they are copied by the imposter. For example see Typekit by Adobe. See also Commercial foundries which allow @font-face embedding.
You can buy some stock photography or other graphics to be used on your website and contact the company that you bought it from if they are copied by the imposter.
You can have some part of your design being changed frequently to make it harder for the fake website to always look exactly the same.
You can find some legal advice on how to successfully sue the imposter for violating your copyright or trademarks.
And last but not least, you can actually inform your users about that problem. You can add a short but visible message to every page of your website advising visitors to watch out for the fake website with a link to more detailed explanation of the problem and ways to distinguish the genuine website from the fake one.
You didn't provide any actual links or say how exactly it is copied, or whether every change that you make to your website is instantly copied to the imposter website or not, so this is the most that I can recommend in those circumstances.