First, the stain is a symptom
A water stain on the ceiling is dried mineral residue left behind by water that came through from above. The stain is not the problem; the water source is. Before any repair, the leak, whether it is a roof, an ice dam, a bathroom above or a pipe, has to be found and stopped. Repairing over an active leak is money thrown away.
What to do first
If the ceiling is actively dripping or bulging, put a bucket down and, if it is safe, relieve the water so it does not spread. Then get the source assessed. Do not paint over the stain hoping it will hide; it will bleed back, and painting seals moisture into the plaster where it can do more damage.
Can the plaster ceiling be saved?
Often, yes. If the water damage is localized and the surrounding plaster is still sound and keyed to the lath, we can remove the failed material, let it dry, rebuild the plaster and blend it back in. If large areas have detached from the lath or the lath itself has failed, replacement of that section is the honest answer. A good plasterer tells you which one you are dealing with, in writing.
How the repair is done right
The sequence matters: stop the water, dry the area fully, remove what has failed, rebuild in coats, then seal so the old stain cannot ghost back through the new finish. Skipping the drying or the sealing is why some repairs stain again within months.