I have a popcorn ceiling with asbestos. I was going to take it down, but saw a program on public boradcasting that outlined the dangers. My research on the topic shows that asbestos is a risk in building materials at least into the early eighties. My house was built in 1972.
The research also showed that painting is a good alternative, but spray instead of roll and make sure that the addition of moisture won't cause the popcorn to flake off. There is a risk that painting will hinder removal efforts, though, and so you may want to be certain first.
Personally, I think testing is your best alternative. It cost me $35 in Tigard, OR. Look up a consultant under Asbestos Consulting in the yellow pages and ask how to safely take a sample. Personally, I felt a ton better wearing a respirator rated for asbestos. You can find them at a Safety Supply store.