Remove snow from around the ice dam using a roof rake. It is a long-handled tool designed specifically for this task. Remove as much as possible from on top of and several feet above or behind the dam.
Create channels through the dam. An already-leaking roof warrants immediate action. Direct a stream of warm water through a hose connected to an indoor tap or temporarily-enabled outdoor spigot on a warm day at the dam's bottom edge. Get the stream as close to the dam as possible and work from the bottom upward to cut a path for drainage. This creates temporary channels that last for a few days.
Dry out the ceiling's interior and assess the damage. Set up fans and increase ventilation so the moisture disappears quickly. Allow the interior to dry.
Scrape uneven areas with a putty knife if the plaster or drywall is loose, bulging or has holes. If this results in small holes, fill them with joint compound; larger holes require replacement drywall or plaster. Sand down repairs with sandpaper until they are smooth to the touch and even with the surrounding ceiling.
Prime the stain with a primer capable of covering stains. Paint over the area with matching paint and a paintbrush.