It has migrated from another patch, and despite pulling it up whenever it shows itself, the roots are widespread, and entwined in the roots of other plants.

I personally would dig up my plants the bamboo has invaded, until I got rid of the bamboo, here is a link for you how to kill bamboo. Hope it helps.
http://www.associatedcontent.com/article/230034/how_to_kill_running_or_clumping_bamboo.html

And I've been trying to grow one for the last two years with no success at all. Strange how plants like one place and hate another.

I would love the black one, I m sure its not so evasive as the other type, as you see as above you can get different varieties clump and running.

Morgana, thank you for your advice, but the shrub border is well established, and expensive to replace, so I am very loathe to dig them up, it would be a major task anyway. I will look at the link you posted, Thank You

Your welcome what a shame about your plants being so established.

Ginelli..thank you for your comment. We have a very acid soil, and it certainly seems to suit the bamboo.

Cut all the stems of the bamboo back to ground. Once the new shoots emerge, wait until mid summer and treat all with glyphosate.

Thank you Fractal...if I'm careful That I only apply the glyphosate to the bamboo, can I be sure it won't kill the other plants?

That is correct. It only enters through green foliage. Being a form of sugar, the plant absorbs it easily through foliage but not through hard stems/bark so you can spray right up to the base of other shrubs/trees. Just keep it off the foliage of other plants.

I would tip (stop) the new stems if they get above 3ft. This will make them branch more quickly making them both easy to manage for spraying and also creating plenty of leaf surface area to get the G. into the plants and its creeping underground rhizomes.

If you have to, leave it even into late summer to get maximum effect.