...is frankly a bit of a bugger. The problem is of course that sound comes from both ends of the instrument, and those ends move around. Usually sound men aim a mic somewhere at the middle of the bellows and hope for the best. A little better is to use two mics, one for each end, but this feels limiting somehow to the player though the sound is improved. In the studio life is much easier as you don't have to cope with ambient sound and can thus use a good condenser mic some way back from the instrument. Use the best mics you can lay your hands on - the timbre of a concertina will defeat cheap mics. My favourite mic for the job is a Rode NT4, which is a stereo condenser that gives you the advantages of stereo miking without worrying about the ends. Plus it sounds really good!
There are manufacturers who make a living from devising mics for awkward instruments on stage - some are listed in section 8. As an example we have used Microvox kit. Their system consists of two close mics which you attach one to each end of the concertina using Velcro. Each mic has a lead which runs into a small box you clip on your belt. From this one single lead runs to the DI box. The advantage is that since both mics are held in proximity to the concertina you can move freely, and the sound quality is not bad too (not up to studio standards but that's not the aim).
It is worth considering in any situation whether you can get away without miking at all. The tone of a concertina is quite penetrating and sometimes in smaller venues where PA is in use we have used mikes for our voices but not for our concertinas.
I asked the concertina maker Colin Dipper whether it would be possible to fit mics permanently within the body of a concertina, but he advised that this would probably have a detrimental effect on the overall sound of the instrument. Having said that, Howard Jones has fitted internal mics to his anglo concertina, apparently without harmful effects, and has put instructions up on the net at http://www.hjcjones.freeserve.co.uk/music/concertina/mike.htm