Weedkiller, and we're coming up to the ideal time. Make sure that you cover any wanted plants, and then spray thoroughly with a glyphosate-base weedkiller. The ground elder will appear to grow better than ever for a few days, but then start to look sickly. You will probably have to apply the weedkiller at least twice.
What's left will have to be dug out by hand - you may well find it best to take up the plants you've planted to get it out of their roots.