First, credit to slawless for putting together the best M3 DIY seatbelt arm repair video on the web. I used it to do my repair. Thanks!
In my case the seatbelt was not too floppy, I didn't have any error codes and the internal "L" shaped transition piece was ok. It appeared the problem for me was that the internal spring had weakened over time. So I simply cut the spring and slightly uncoiled it to create increased tension. If this is your problem, try this free fix (EDIT- This fix worked for about 250 vehicle starts for me, which could be months for some people or years for others):
1. Remove the arm following slawless's video above.
2. Check if the "L" shaped metal transition piece is broken or not. See this edited photo sourced from Josey McWales's post on E90Post.com:
2.a. If the metal piece is broken, try Josey McWales's simple, nearly free fix here:
http://www.e90post.com/forums/showthread.php?p=18011868
2.b. If the "L" shaped piece is not broken it should look like this (note: large spring in photo already removed):
3. Remove the larger of the two springs here by pulling up towards you on either end. Now you will need to cut the spring. I used a pair of mini wire cutters to do the job. Takes some effort, but they will do it:
4. The cut spring will look something like this:
5. If you can't fit the newly shortened spring back in place because it is too tight, unwind enough of the coil to just barely be able to fit the spring back in place. Here is where you will want to unwind it:
6. Now it should be fixed! Note it took about 10 extensions and retractions before the seatbelt arm "settled in" and started grabbing the belt each time. Since the first 10 operations, it has now been nearly flawless for the last 200+ times I've used it failing to deliver the belt only once.
7. This free fix is worth a try if you are going to replace (and throw away) the part anyways like I was planning to. The new part appears to have a different type of metal "L" shaped piece that looks like it won't break apart like the original design. Here is a photo of the revised design with the new piece (driver's side part number for M3 only 72117334637). I don't know my metals very well, but the old one looked like some sort of cheap "pot metal" or die-cast zinc allowing it to easily crack. Then new one looks like it will bend before it breaks. Maybe stainless steel or something similar:
Good luck!