liftman’s right.
There is a chemical, Ammonium sulfamate, which used to be recomended for accelerating the decomposition. It’s sold as ROOT OUT – but they don’t recomend it for that any more. Don’t know why.
I’d go for digging it up, but if it looks like you can’t chase the individual roots out without destroying the garden, chop ’em off and get the bulk out, and leave a bit of root out on the remnants.
It is a job that just seems to grow as you discover more roots underground, so if a pro will do it or you can use a mini-digger then that’s a good shout. i guess it depends how big the stump is.
You don’t want to encourage honey fungus, it’s a pain to get rid of once it establishes, lives for years and kills your other trees and plants (but apparently tastes sweet).
http://en.wikipedia.org/wiki/Ammonium_sulfamate