If you're familiar with the work of aging researcher Michael Rose, you'll know that he uses "immortality" to mean no increase in mortality rate with advancing time. He demonstrated that mortality rates in flies, for example, stop rising after a certain age. This is counterintuitive for most people, given our experience with the world - we expect the chance of death on any given day to continue to increase as age-related degeneration piles up.
The flies all still die, of course, because the odds come up sooner or later, but they don't get any more likely to die per unit time whilst in this "immortal phase." You can find a more detailed explanation in Rose's essay in the online version of The Scientific Conquest of Death - it's the first essay in the book.
Biological "immortals" will often die, just not because of a systematic, endogenous, ineluctable process of self-destruction. Death is not aging. Biological immortality is not freedom from death. Instead, the demonstration of immortality requires the finding that rates of survival and reproduction do not show aging.
In caged insects, kept under good conditions, mortality rates stop increasing in late life. The new facts of death reveal three phases of mortality: juvenile, aging, and late life. In the juvenile period, mortality rates do not show sustained increases. In the aging phase, mortality rates increase rapidly. In the third phase of life, mortality rates are roughly constant, though they tend to maintain a very high level. Organisms that reach the third phase can be said to be biological immortal, in that they no longer age.
This all sprang to mind when I noticed a paper on cancer rates in the old:
Increased age is regularly linked with heightened cancer risk, but recent research suggests a flattening around age 80. We report that, independent of cancer site or time period, most incidence rates decrease in the more elderly and drop to or toward zero near the ceiling of human life span. ... Almost all cancers peak at age approximately 80. Generally, it seems that centenarians are asymptomatic or untargeted by cancers. We suggest that the best available justification for this pattern of incidence is a link between increased senescence and decreased proliferative potential among cancers. Then, thus far, as senescence may be a carcinogen, it might also be considered an anticarcinogen in the elderly.
Your guess is as good as mine as to the mechanism by which cancer rates fall after 80, though the centenarian mention is probably a red herring. Centenarians tend to suffer less of everything bad, being more healthy and active than their peers at every age, as a result of (to some degree) genetic luck and (more influentially) lifestyle choices like exercise and light calorie restriction. That every cancer type is affected probably points to something global, such as the immune system, but nothing plausible springs to mind.
In any case, we can speculate that this is one component of Rose's late life immortality, as demonstrated in humans. Less cancer means a decreasing contribution to mortality rate due to death by cancer. Does human mortality rate in fact flatten out in old age? I believe that remains debated but plausible, lacking the irrefutable sort of statistical data one can produce with flies in the laboratory.