REGULATOR In water near freezing point, CE markings on regulators are meaningless. The standard requires testing only down to 4C; a reasonable limit for normal recreational use, but tropical compared to Antarctic conditions.
Another standards consideration is that a very expensive regulator testing machine could ice up and break!
I asked around among divers who should know. The British Antarctic Survey uses Poseidon Cyklon regulators and says it has no problems with first-stage ice-up, although the second stages do occasionally freeze and free-flow in the Antarctic winter (BAS divers are the only ones who work through the winter).
Divers in Scandinavia who regularly dive in near-zero temperature gravitate towards either Poseidon or Apeks regulators. When I mentioned brands with piston first stages, they just laughed. My preference is for an old Conshelf 14.
Whichever regulator you use, the best defence against freezing is to keep it absolutely dry until you get in the water. Some divers also reduce the inter-stage pressure by 2 or 3 bar, trading an increased work of breathing for less expansion and hence less cooling in the second stage.
Yvette Cardozo says she went to Antarctica toting Sherwoods Maximus and Blizzard regulators and liked the way they worked. I used the Maximus as my primary and the Blizzard as my safe second. The Maximus breathed easy, even when I was sucking air like a marathon runner. At no time during eight dives in water hovering around -1C did either regulator freeze or free-flow, including the time I purposely purged both second stages in shallow water to try to simulate a free-flow.
In studies done in Antarctica by the US Antarctic Program, the Maximus outperformed the Blizzard plus 16 other regulators, she says. USAP then adopted the Maximus as its official regulator, and over the following two years and 2000 dives, reported no failures.

DRYSUIT Even the smallest leak can make a major difference to warmth, so get up to date with puncture repairs and ensure that seals, zip and boots are all in good condition.
My contacts are split between neoprene and membrane, with those into longer, more technical dives preferring membrane suits made by Polar Safety (often badged as Dive-Rite) with an integrated hood. I usually prefer a neoprene drysuit, but in really cold water switch to a membrane with a Weezle undersuit and an extra layer of thermals underneath. (Yvette Cardozo didnt wear a Weezle but says: After watching one guy actually sweat on a dive, I ordered one as soon as I got back!)
The BAS currently uses full thickness commercial-grade neoprene suits from Northern Diver with flectalon undersuits.
Hood Hoods integrated with drysuits are very popular in Scandinavia. My preferred item is a full-face hood with a hole cut for my mask and a slot through which a regulator can poke. Its almost as warm as a full-face mask, requires no special training and is a lot less hassle. However, it isnt practical for a full-face hood to be integrated with a drysuit, and it does make me look like Hannibal Lecter.
The BAS uses standard neoprene hoods with no modifications and no full-face masks.

GLOVES Dry gloves may be extremely popular with extreme coldwater divers, though they are not used by the BAS as they are not robust enough for dives involving a lot of hand work.
Next on the list after dry gloves are three-finger mitts of 5mm or 7mm neoprene. The BAS uses full mitts in 7mm neoprene.
There are definite tricks to tasks such as fitting a mask, manipulating camera controls, tightening straps and all the other things divers need to do without thinking about it while wearing dry gloves or mitts. You dont want to spend the first few dives of a very expensive trip learning, so a few pool and quarry dives before departure are advisable.

MASK A mask has to both fit your face and match the hole in your hood to leave minimal or no bare skin exposed. The shock of a mask flood or, in the worst case, having to do a mask removal and re-fit in zero-temperature water is painful. Nevertheless, it is something with which you may have to contend, especially if you are wearing a full-face mask and a regulator freezes.
The best opportunity to practise is at the bottom of a UK quarry at the end of January, when water temperatures can get down to 2C.

BC Make sure that your BC fits comfortably on top of your drysuit and all the extra underclothes you may be wearing. BC and drysuit feeds can freeze in the same way that regulators do. Get your weight right and eliminate unnecessary use of feeds during the dive. Practice emergency disconnects with gloves on.

DIVE COMPUTERS Battery performance drops in low temperatures. Some types perform better than others, but with most dive computers you have no choice as to which battery to use. The best you can do is start a trip with a fresh battery.

CAMERA & FLASHES I once had a partial camera flood that I traced back to a small crystal of ice forming on the main housing O-ring from a drop of moisture. Whenever your camera is open, make sure the O-ring is dry before closing it again.
When you load a camera in a warm humid atmosphere, then take it diving in cold water, there is a good chance that fog will form inside the lens or housing. This may not clear until 30 minutes into a dive. One trick is to leave the camera in a bucket of water for half an hour before the dive, but that wont work if the fog freezes inside the housing. And chilling the batteries down may be counterproductive.
I generally try to load cameras well before a dive, and wedge a small sachet of silica gel in a corner of the housing.
I use nickel-metal hydride (ni-mh) batteries in my flashes and have never had any problems arising from cold flash batteries. Once I start taking pictures, the batteries generate enough heat to keep working happily.
With camera batteries, the current drain is so low that no heat is generated. I have found that while cheap alkaline batteries work well most of the time, they occasionally lead to cameras seizing up in cold water. My current coldwater preference is for the new Duracell Ultra batteries in the camera body.
I have had no problems with film behaviour in cold conditions. The British Antarctic Survey says it makes no special allowances for the effects of cold on film speed.
When bringing a dry camera indoors, the BAS recommends placing it inside a large plastic bag so that condensation forms on the inside of the bag rather than on the outside of the camera.

REBREATHERS A rebreather offers the attraction of breathing warm moist air. Weighed against this are reduced scrubber efficiency and the prospect of electronics not behaving in the cold.
A neoprene glove fitted to the scrubber canister helps to keep it warm. Pre-breathing before a dive to get the scrubber warmed up is always a good idea, but especially important in very cold conditions.
With warm humid air inside the breathing loop and icy water outside, far more condensation than usual will form within the rebreather. This is unlikely to freeze while its in use, but could do so when equipment is removed after a dive and left to stand in the cold.
Between dives it is best to get a rebreather inside and thoroughly drain and dry it. Change the scrubber well inside the normal interval.

CONSIDER THE AIR TEMPERATURE While seawater temperatures can drop to -2C, air can be much colder. Regulator hoses become fragile and easily gouged by the metal crimps at the ends. Some plastics become brittle. Batteries can get so cold that they stop working.
Between dives, nothing will dry if left outside. Moisture will freeze then slowly sublime, but not in any useful timescale. I have known divers having to crack the ice on their diving suits and undersuits to get them flexible enough to put on.
You need to get all your equipment inside and thoroughly dry between dives.