Well, you don’t lose them: you return them to their rightful owners. 
Brazil has been through this a couple of times, our tribespeople had some interesting anthropological structure by which they did not perceive or act as if our culture posed a threat to theirs (instead, syncretism sprouted), but they were very resistant to being enslaved or killed or to having their lands taken.
During the Empire and the Old Republic, our armies devastated most of the tribespeople, but the New Republic has, starting with the constitution, tried to guarantee the tribespeople relative independence — that is, we’ll give you more and more lands, and we’ll let you choose how much into civilization you’ll be, furthermore, you can follow your own law as long as you don’t violate our law, in our space.
Of course, this sounds more beautiful than it works in real life. There are a couple of issues, but quite honestly, it has been working better than nothing. In fact, some tribes in the Centre-West and the Amazon areas remain untouched: we know about them, but they remain clueless to the existence of a whole civilization around them.
Like the news said: it’s conforming to the international treaties of self-determination, and according to the supreme law of the US. Their declaration of independence is legit.