It seems that the plan to recreate woolly mammoths is already in progress. Not just talking and planning, but actual progress. Three genes from the mammoth have already been migrated into elephant cells. In three years, we could have a mammoth-elephant hybrid embryo ready to be implanted into an Asian elephant uterus.
I emailed George Church (the scientist doing the project) to see if it was true. And it was indeed true. You can see his email in the thumbnail.
Instead of trying to recreate a full mammoth genome (pretty much impossible), he is instead creating a mammoth-elephant hybrid genome, which is much easier. George Church's TED talk explains this further. The genome-editing method he uses is known as CRISPR.