The Organic LAB

This is an amazing restaurant, but do note that it is high-end. Food is usually expensive in Copenhagen but The Organic Lab is a bit more expensive than usual - it's also very modern fancy-restaurant-ish. By expensive I mean that a full three-course menu costs 323 DKK, which is around 43.5 Euro.
Portions might look small, but you do get full!
This restaurant is fully vegan, but I've brought 3 meat-eaters, and they absolutely loved it. All drinks, wine, etc. are fully vegan as well, and the staff is nice and definitely know what vegan is.

The restaurant /is/, as many vegan places in Copenhagen, very small. You sit on very high stools at slim tables, so even though the food is high-end, it might not be the best place, if you're looking to have a really relaxing and fancy night out.
I would recommend booking a table in advance, in the weekend.

Updated from previous review on 2017-02-28

Pleasant little new Nordic style vegan restuarant in the Nyhavn area. The evening menu had two starters and two mains plus dessert (descriptions were in both Danish and English). I chose the baked celeriac which was truly delicious and everything was beautifully presented. The best place I ate on this visit to Copenhagen.

Amazing food, great flavours, perfect presentation and really nice staff. So worth it!

This place is a gem. Really, really great tasting food served with charm. Do yourself a favor and visit this gem of a vegan food heaven in the middle of Copenhagen.

I wished that there were more places like this, food prepared with a lot consideration to flavour and texture and great service in an interior that inspire you with artwork display. Cozy and fancy at the same time😊