I'd dig out the stump and as much as possible of the roots, then treat any regrowth from remaining suckers with weedkiller. There is a powerful brushwood killer around but I've never used it, someone who has may suggest how to use it - if appropriate here.

SBK, you drill holes in the stump, fill them with the SBK and then cover the holes with something to protect the wildlife. Bay trees are known for regrowing from the stump.

As Moongrower says, you need to poison the stump to kill it off, if its too large to dig out, and it sounds like it is. Use a power drill with a medium to large wood bit, drill the holes about quarter to half inch deep if its a large stump, in particular near the outside edge, about an inch in, till you have a ring of holes about 2 inches apart, then about four into the middle. Fill with the SBK, cover and leave. Don't spill the SBK anywhere, don't get it on yourself either - its like water, so not easy to handle without getting it everywhere. If any regrowth next Spring, re-drill the live area and repeat.