This is taken directly from the BBC Gardening message board - If you are not happy with this answer it might be an idea to put your question there, as they are incredibly knowledgeable.
"Ivy propagates itself by producing roots wherever a stem touches the soil, thereby producing a new plant. Cutting or ripping away stems casually has little effect, the portion of stem with roots must be removed too or it will regrow, multipying the problem. Fortunately unlike bindweed, ivy does not usually regrow from underground roots, so it is not nessesary to dig out every bit of root. Neither will it grow again from isolated bits of stem on walls, fences etc - these can be left to die off.
You will almost certainly not get it all in one go no matter how thourough you are. It is best to do as much as you can and then leave it - any remaining live material will soon regrow and you can then remove these too. With a realy bad infestation it may take severall gos to irradicate the ivy. But with perseverance you will get there".