Then I stumbled upon SP Upgrade and it worked so fine! Absolutely uncomplicated, same IDs, including monitoring/log (was very helpful for reviewing the migration, because I have a very large website).
The migration interrupted for 3-4 times (due to my server), but I could continue migration from the point where it stopped.
I am so happy! Thanks :)
The support is excellent, too. The Support Staff spent several hours on investigating a problem (it turned out that it had nothing to do with the JV-Relatives), even though it was easter holidays (!).
Thanks a lot for your great work and great support!