Fertility would stabilize around replacement rate once life expectancy stabilized too. The number of children in a population would still go down though, as for any one woman's 1,000 year lifetime, there would be two children.

Unless, of course, you mean that people who live to 1,000 would be children (immature) until they reached 180 years of age. I.e., just scale our perception of time along with our life expectancy.

But merely scaling time isn't interesting. People who want to live forever want to live forever as adults.