Obviously you can't dive in freezing water--it'd be ice if it were freezing. However, you can induce the water inside the ambient pressure part of the first stage to freeze in cold water. When the air in the tank leaves the cylinder, it necessarily has to expand. One of the byproducts of air expansion is cooling. If the air expands rapidly enough, it'll cool the water in the ambient part of the first stage enough for ice to form. This ice will restrict movement of the piston or diaphragm (depending on type of first stage), causing it to be unable to close up after you inhale, and you will have a free flow. This freezeup is more likely the colder the water you're diving in.
Cold-water divers routinely use a "sealed" regulator to eliminate freezups. The first stages are sealed to prevent water from getting inside them. Instead, they use either a silicon oil-based medium or a plastic piston behind a watertight seal to transmit depth information to the diaphragm. Sealed piston regs use Christo-Lube as a sealing medium--it's just plain nasty to work with!
Cold-water second stages usually have a lot of metal in them, either the whole case or the breathing tube. Metal transmits heat much better than does plastic. By being metal, they transmit heat from the water (even when it's cold, water has heat in it) to the air in an effort to keep any moisture inside the second stage from freezing.
Hope this helps some.