As far as I know, the problem is modern cassettes being to short (really the spools in the cassettes). Any which way, it is easily fixed. There was a thread a while back on how to fix it. Someone tried mounting a conical spring on the baseplate to push the cassette in to place. Reportedly worked fine.
I myself have used paper to shim the locking mechanism where the baseplate meets the cassette (in a Leica II and a III (my IIIc didn't need it)). You'll have to waste a film to do it. Load as usual. Wind on several frames to ensure everything has settled. Open shutter on T or B, and with the lens off, carefully (don't blame me if you muck anything up by sticking a pen in your Leicas people!) draw lines, following the edges of the frame aperture. Take the film out to see how tilted the frame is, wind it back in to the cassette. Use a thicker paper shim and redo using the same film until your frames are straight.
Was that understandable?
As far as I can remember (neither camera is here right now so I can't have a look) I shimmed the III's baseplate the proper, and harder, way (dismantling the locking mechanism, inserting paper shim under a metal shim), but doing the II was a bit of a quick-and-dirty job. I just cut a piece of stiff paper, sort of mushroom-shaped (you'll know what I mean when you look at the mechanism) with a center hole for the screw, a little too big (to make a good push-fit). I expected it to wear out quickly but it's still going strong after 200+ rolls.
YMMV.
Hope it helps...