Reregisering car in new commune?

I have just moved from Ixelles to Woluwe-SL comune and have also registered my presence with the new comune. My question is now: Do I also have to change the registration of my car with the DIV or through the insurance agent, who did the registration formalities for me?

The nice people at the commune couldn't tell me, but said that it would take 5 weeks to get my ID card updated, so I might not be able to do it before that. I can't find anything on the DIV homepage and they don't answer the phone (at least not within one hour of waiting).

Any experience, any advice.