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After adding the new card, the "Networking" tab has a fresh new look, and now all kinds of new stuff is configurable:
After enabling IPv6 I settled for an auto configured IPv6 address. This incorporated the MAC address but this did not seem an issue.
I tried to disable IPv4 but there was no way of doing that. I configured an IPv4 address in the 169.254/16 range to 'kill' IPv4.
After confirming it actually worked I also removed the A record from the DNS.
Most of the office computers did not even notice the change, however some computers would not print any more. The only resolution was to remove and add the printer again. I suspect some sort of caching issue.