Remove as much furniture as possible from the room. Cover the remaining furniture and flooring with plastic. Secure the plastic to the walls with painter's tape so it doesn't slide around. Lay newspapers on top of the plastic to make it less slippery.
Mix equal parts sudsy ammonia and water in a spray bottle and spray the ceiling. Work in small sections, about 4 feet by 8 feet. Let the solution sit on the ceiling until the ceiling starts to soften and bubble.
Scrape the softened stippling off the ceiling with a ceiling-texture scraper. This tool has a bag attached to catch all the stippling as you scrape it off. Work carefully to avoid gouging the ceiling with the tool.
Continue spraying the ceiling and scraping off the stippling until it's all been removed.
Sand the entire ceiling with 100-grit sandpaper to smooth it and prepare it for painting. Dust it with a tack cloth or vacuum it to remove the sanding dust.