Ground elder can be difficult to eradicate as it has brittle roots which break up easily and produce shoots from the smallest piece left behind. Even so its still a good idea to dig up as much of the root as possible with a garden fork. With patience sieve the soil to remove smaller pieces of the root! Also use weed killers if the ground elder troublesome. Try leaving the weed to grow (to increase the leaf surface area) and then treat it in mid-summer with a glyphosate-based weed killer such as Roundup, which kills weeds right to the root. If the ground-elder is near any ornamental plants, cover them with plastic sheeting whilst spraying the ground elder, as glyphosate will kill any plant it comes into contact with. It is also beneficial to spray the weed killer in the evening as more of it will be absorbed by the plant than during the day.