Thanks for your suggestion Idunham1. The problem is that the new skeleton doesn't have the same number of joints, nor the same names.
So I ended up re-weighting the skin to the new skeleton. But I wish there were a tool to automatize this process. Something that would look at the bounding boxes of the old joints, define which new joints match them best, then re-apply the weights based on world-space vertex coordinates.
I mean Maya has a way to replace a skinned geometry, and it should also have a way to replace the skeleton of a skinned geometry...