Stabilization is key to the longevity of drill bits.
Power drilling is problematic , the bit catches and its over.
Not many Dremels or other small hand drilling machines go slow enough for the work.
UMM makes a smaller chuck that can be put in a pin vice or held by hand:
http://umm-usa.com/onlinestore/product_info.php?cPath=21_161&products_id=3338As with all of their stuff its very well made and versatile.
I also have modified small wood handle pin vice, cutting down the wood handle:
This may be a better holder as it has less weight than a pin vice , the weight of the holder alone being enough to easily break the drill bit.
Don't be afraid to modify or make your own tools , if it works then that helps the problem solving.
As for technique ; the 'bite' of the bit , that is to say when it has cut into the material , has enough tension to break the bits. This is where I think your problem may be Gaz. I would suggest the smallest rotation in the cutting direction , no more than half a turn , then reverse rotation with the tiniest pull out , this will minimise the chip and cut it off with less tension. I set up a back and forth motion with my fingers , practise this with your fingers , try to keep the motion even and slow , back and forth. Also I will pull the drill in an outward direction as I rotate in the cutting direction , this minimises the cut.Stabilize the parts , the drill and holder and yourself as well.
Takes some skill and practise , but that's the essence of modelling.
And taking into consideration that it is unadvisaeble to say ANYTHING bad on the internet , I will state here that purchasing cheap pin vices from ebay or otherwise can be counter productive. I bought a set from a UK SUPPLIER and found them to be very very poor in quality. The drill will chuck up ok but upon examination the quadrants that recieve and hold the bit are machined inexpertly , therefore the drill is OFF-CENTER .
Watch out for this and buy from machine shop supply or model tool suppliers.