Well I can now confirm that it does depend on the make of Rechargable battery you buy that makes a difference to whether they work in the Magic Mouse or not.
Last night I took a gamble and bought some Duracell Rechargables. I didn't have my glasses for close up work but I was able to compare them right there in the shop with the black alkaline batteries that I know work fine. The cap in the end looked ok though - about the same size. They weren't cheap so it was a bit of a gamble as I don't have many other uses for them right now.
First of all, they only took an hour or so to charge up. That was my first surprise. However when I put them in Magic Mouse - lo! - they work perfectly!
I took another photo that shows all three batteries together.
http://dl.dropbox.com/u/11373233/MacDiscussions/Magic%20Mouse%20folder/Result.jp g
The new Duracells (on the right) are clearly as large at the + end as the Alkline cell (in the middle). It is definitely larger than the previous set of rechargables (on the left).
Last night when I first used them they were at 94% charge. This afternoon (3pm) they are at 77% charge according to the Mouse Prefs Panel. I don't know if that is a lot of charge to lose in less than 24 hours but a great dealof use has beenmade in that time. Doesn't matter though - I can recharge them, right?
Anyway the ones I found that work in my Magic Mouse are Duracell 'Rechargable Accu' NiMH, 2000 mAh/1.2V. They also claim to retain 80% of their charge after one year. So it may well be worth charge up the next set in readiness.
So I'm pleased to be able to report a good result.