Keyword and Rank-Stealing

Hello,
I noticed that a website steals my rankings.
It has exactly the same title, description, keywords and URL pattern but the domain is different.
I know it sounds a little bit strange and hard to believe but it seems like that page takes over the whole Rank(aka position in Google) and my Rank(with my domain) does not appear any more. Not even on page 25. It looks like this mysterious page replaces all my domains (and all listed ranking as well). When I try to visit this new page I receive a https-error.

Example:
Phrase: hello World

My old Ranking ast position3:

Hello world from France
click her to learn more about france...
franceguide.com/hello/world


My new Ranking at position 3:

Hello world from France
click her to learn more about france...
motorconstruct.org/hello/world


As I said my old ranking will not be ranked down, it's disappears completely or will be replaced.

Did anyone saw such strange behaviour in the big G?

P.S. I found out that my domain and the "thief-domain" is hosted on the same IP.
Furthermore I found out that "thief-domain" is redirecting the traffic back to my page (god knows why). But to the https (self-signed) version of my page. To prevent that all users are seeing the Https-warning, I placed a .htaccess redirect to my normal http version, now everything works fine.
But in Google is still the "Thief-Domain" listed instead of mine.

I had this problem when I had malware on my website.. And google bots were redirected to the hacker's website.. then my website was replaced with his.. did you check for that ?

I checked everything. I have a script which computes the md5 of all php, html and htaccess-files and everything is the same since I moved to that webhosting provider in august this year.
My website stays the same.

I would like to report this to the Google Webspam Team, due this technique is very sophisticated and dangerous. OK It requires that the IP of both websites (victim and attacker) is the same nevertheless for shared-hosting users this could be risky enough even when their website was not hacked.

P.S.
May as you know even when your .htaccess files are unchanged/clean, the header-infos can be different due a webserver like Apache can manipulate such header as well. So even when your .htaccess files are clear a hacked webserver could manipulate the http-headers.

So I did a http-header scan by visiting my page with activated "Live-Http-Headers Plugin" in order to find some permanent redirects to the attacker page but everything was clean (no redirects, URL change notifications what so ever). Than I changed my User-Agent from Firefox to Googlebot but the headers stays the same.

Slowly but surely even my knowledge is exhausted.