Spirochete:
Regarding the immortality of single-celled microbes, it doesn't happen, and here's why:
When you grow bacteria up in batch culture (that is to say, in a flask, with a pre-defined medium and set nutrient level, seal the system, don't add anything,) the bacteria go through five distinct phases:
1) The lag phase, where the microbes adjust their protein expression to the nutrient composition of the provided medium,
2) The exponential growth phase (also for some diabolical reason called log phase, but I hate this :-( ), where the microbes behave as if they were growing in an infinite system, without bound,
3) The stationary phase, when the microbial population begins to be starved for nutrients, as it is too large to be sustained on those nutrients provided,
4) The death phase, where around 99% of the population dies for the same reason, and finally
5) Long-term stationary phase, where the primary nutrient consists of dead bacterial biomass.
Now, there exist dyes that stain only actively dividing cells. (I forget what they're called at the moment.) If you use these dyes to stain a culture in the exponential growth phase, most of them stain positive - this makes sense, since during exponential growth phase, the cells are dividing as fast as they can. However, once you slide into stationary phase, the stain starts showing up negative, and in death phase it's obviously even worse.
But the tricky bit is that if you stain a culture in long-term stationary phase, you find that most of the cells that are alive are dividing again! In fact, if you take the same batch culture, and plate out samples from when it is 5 days old, and again when it is 10 days old, and 20 days old, etc., and you look at their most basic properties like color, shape, and size, they are all different. Additionally, if you inoculate that 20-day-old culture into the 5-day-old culture, the 20-day-old culture will take over the 5-day-old culture; it will reproduce more successfully and drive the 5-day-old culture into extinction. It's been shown that the later cultures are more efficient at taking up amino acids from the environment (read: the remnants of dead cells) than the earlier cultures are.
The moral of this story is that long-term stationary phase is only "stationary" with respect to total cell concentration, not cell division, growth, or death. What appears to be stationary phase in terms of cell density is really a succession of individual populations, each out-competing the one before it. Strains that are advantaged in such environments are said to exhibit the Growth Advantage in Stationary Phase (GASP) phenotype; you can Google-scholar the topic for more info.