Its most likely Curl grabbing your pages from the remote server, its not a parked domain I checked and your httpd.cnf doesn't allow it.
I would 1st check if the remote site caches pages, goto the clone site and view loads of pages while checking recent visitor IP's in cpanel. if you see may requests from a IP that isn't you own ban it from your site.
Failing that disable hotlinking using cpanel or a wordpress plugin.
They cache the pages.
I sent an email to the owners of the site (if you go to Zarephath Health Center you will see it is a health center site).
I believe the owners may not know they are being hijacked by someone and all this is added to their site without their knowledge.
Will hopefully get an answer from them.
Nothing in cpanel, as right now, strange that only 40 visitors, but 9,000+ page views!
Oh well, whatever method is used, they capture my new posting like a minute after I post!
9000+ page views would indicate the clone site server is connecting to your site on a cron, or not using caching.
If you look at the access logs from your cpanel it shouldn't be to hard to find the offending IP that needs banning, but again you should see something in recent visitors page.
They are likely scraping your site directly via a proxy service (so the IP is changing around), so not caring about the RSS feed or copyright law.
Your best option is to get the website host to remove the (PERL) script from the infected server IMO.
Anyway, the duplicate site should hopefully be penalized in the SERPs automatically.
The problem is that the duplicate sites are not always penalized. Google frequently screws that up and penalizes the wrong site.
According to something Matt Cutts once said, Google assumes that the site or page with the most backlinks is the original. They don't do it on the basis of which page content was found first.
It looks like the hosting was hijacked or a script dumped onto that site.
Have you tried to change a few words on a page to see if they change on the clone site?
What is strange is that it appears to use the zhcenter.org domain, but that domain still shows the health care site.
"Democracy is two wolves and a lamb voting on what to have for lunch. Liberty is a well-armed lamb contesting the vote." -- Benjamin Franklin
But the duplicate site is deeper in the URL so I would not worry about it (unless it was my site LOL). Well, anyway, put pressure on the Hosting firm to delete the script (zh.pl or whatever) and scan their servers for a similar infection.