Looks like that's not going to happen. From Slate:
A somewhat more arcane milestone, meanwhile, generated no media coverage at all: It took humankind 13 years to add its 7 billionth. That’s longer than the 12 years it took to add the 6 billionth—the first time in human history that interval had grown. (The 2 billionth, 3 billionth, 4 billionth, and 5 billionth took 123, 33, 14, and 13 years, respectively.) In other words, the rate of global population growth has slowed. And it’s expected to keep slowing. Indeed, according to experts’ best estimates, the total population of Earth will stop growing within the lifespan of people alive today.We knew affluent societies stop having children. Europe has been facing this problem for years, as has Japan. Reading the article, however, there's a huge shocker. We anticipated that less developed populations which tend to have many children, would simply expand to fill the void. Europe's "native" population would be replaced by immigrants, as would happen in the U.S. once we actually fell below replacement rate.
And then it will fall.
That isn't happening. Third world birth rates are falling, too.
It would take us many years to die out as a species, but this is a fascinating development that will make for interesting study for years to come.