Contacting the website's hosting provider is a very good way, but you are going to have to show that your site was up there first. Unfortunately (or fortunately depending on your perspective) most hosting providers put the burden of proof on the plaintiff (in this case you).

One way to do that is using the wayback machine, Internet Archive: Wayback Machine which may be able to show that it has an archived copy of your site up and running prior to any archived version of the other site.

Good luck.