The best way is to dig over the ground thoroughly. It is shallow rootrd and the roots will lift easily from loose soil. A Glyphosphate weedkiller such as Roundup is the second best option. If it is growing among garden plants then you could put some of the weedkiller in a bowl and apply it to individual leaves with your fingers.

Dig it up and make shore you get all the roots, any part of root left will regrow again, do not put ground elder in recycle bins

And what you can't get rid of can be used as a pot herb or a remedy for gout! It used to be grown in monastery physic gardens for that purpose. I did try it and have to say I wasn't over excited by the taste!

Bulbaholic says
"If it is growing among garden plants then you could put some of the weedkiller in a bowl and apply it to individual leaves with your fingers."

Please don't do this with your bare hands. You need to wear gloves if you're going to do this - make sure they're new with no holes in. You need to protect your skin from exposure to all weedkillers including Roundup, and wash carefully after using it.

You could make a solution of Roundup, thicken it with a little wallpaper paste and apply with a paintbrush.

By the way, my OH informs me that some organophosphates (and Roundup is an organophosphate) can get through rubber gloves. The safety data sheet from Monsanto advises chemical resistant gloves for prolonged exposure.

Dig it out as much as you can. Spray with Round Up as soon as you see any regrowth.

Also have a look at my profile page where I describe what worked for me .

In addition to some of the suggestions above re using round up round the base of some plants, Poke the elder leaves into a small hole in the bottom of a carrier bag, then spray into the bag, leave the bag in situ for a couple of hours. this eliviates the problem of rubber gloves etc. You might end up with a lot of bags over the problem, but it's better than touching the stuff.

I gave up trying to dig out elder when I found roots were a good 2' down in an old garden I used to own.