Proxy redirect - only how?

I need a redirect without changed headers - users should not see the change

[MartinGaedeke]

Hi everyone,

I want to move my site to another server and I want to do it without any downtime.
So I was thinking about a proxy redirect form
www.old.com/.*
to
www.new.com/.*

But one thing is very important for me: the headers are not allowed to change. Neither visitor nor search engines should notice the change!
Unfortunately I am only allowed to use .htaccess :-(

Until now I tried the following two methods:
ProxyPass / [new.com...]
That didn't work and I guess it's because I can't use it in .htaccess (?).

RewriteRule ^.*$ [new.com%{REQUEST_URI}...] [P]
Sadly this creates some new Headers, e.g. "X-Cache". I tried to remove them but it only seems to work with Headers that I manually create myself:

Header add TestHeader "test"
Header unset TestHeader
Header unset X-Cache

"TestHeader" will be removed but "X-Cache" still exists! :-(

So is there another way to accomplish this task? Or did I do something wrong?

Thank you very much for your help!
Eddie