A person, or persons, would decide this was a group no one would care about the annihilation of, and would wipe them out and make off with the vast amounts of wealth.
I would just like to point out that this can be considered a ridiculous statement given that it take a considerably large group of people to overthrow an entire civilization in the current global theater. It's not impossible, but if it was widespread enough that an entire "society" was doing this, it wouldn't be something that would simply be changed by some sort of "cleansing" without immense losses on behalf of the "rebels," especially when you consider the technological advancements that an immortal society would be able to achieve (longitudinal studies suddenly become far more effective to the point that many more psychological and medical disorders and illnesses are eradicated or "subdued," etc.). Such a society would also be less concerned with death (since it isn't a natural part of their life), so I doubt that war would be seen as as big of a deal to them (they would be willing to deploy more soldiers, fire more missiles, etc.) and their massive surplus of people would mean they could easily afford to have people die. Also, that massive amount of people means they would have much higher productivity as a society, be capable of developing new technologies (including military technologies) at an accelerated pace, and be able to produce massive amounts of those new technologies.
In short, trying to stop an established society of immortals
is going to be a very bad idea (in terms of reciprocal losses, or simply chances of failure) in most cases.
Also, I stand by my earlier suggestion (operating under the hypothetical assumption that this would be a viable route to immortality): use stem cells to grow hearts.