What I wrote above was specific to a second stage ice-up and free flow. I have a Sherwood Magnum Blizzard regulator from the later 1980s that has two copper vanes inside the mouthpiece which are screwed into the top of the second stage valve, which transfers the heat from the diver's breath to the mechanism, which significantly reduces the potential for second stage freeze up.
The Magnum Blizzard also has a unique first stage, which has a modification to the first stage which allows pressure to be transmitted to the piston, but excludes water. So water never gets into the first stage, thereby eliminating the possibility of water freezing inside the first stage.
Other regulators have displaced the water in the first stage near the piston by silicone grease. Care must be taken to ensure that the silicone stays on the outside, displacing water, and doesn't get inside to the mechanism, where it can be transmitted through the intermediate stage/hose system to the second stage, where the possibility of inhaling the silicone grease present themselves.
So we really have two separate situations that can cause a freeze up and subsequent free flow of air.
SeaRat