Left ledge method
In this method you fall to the near left ledge from the V without bouncing off. From there you can move along and drop down at a safe spot of your choosing. Suitable only for NTSC/PC.
Get to the bottom of the V and face left (up the battlefield). Run up the side of the V a short way then leave it, practically parallel to the cliff face. As you fall, curve around to stay close to the cliff. It bulges out somewhat and you don't want to hit that and get nudged away, else you won't make it to the right spot on the ledge. You can curve more sharply once the cliff is going inwards.
You'll need to land in a quite specific area along the ledge to avoid dying on impact or slipping off, so it may take some experimenting. Don't crouch as you land, but do try to stay on. If you're facing the cliff at this point, that means trying to run forwards to stop from sliding off (I'm not sure if it's equally viable to land facing outwards and then stay on by running backwards). It's possible to survive the ledge impact without even a flicker of your shield.
From the ledge you can move along and drop down at a safe spot of your choosing, as described elsewhere.
If you want to get a good idea of exactly where along the ledge you need to land, have a look at cocopjojo's AotCR Descent movie or slYnki's tutorial. Look at the dark markings or whatever, and try to identify them when doing the fall yourself.
To make things as easy as possible for repeat attempts, get a delayed checkpoint when ready to start your run off the V.
Difficulty and system differences
I suspect this method is impossible for PAL, due to insufficient traction when you hit the ledge. I always slip off. I'm not even close to being able to stay on. I've seen other opinions expressing the same conviction about PAL (see here or here or here, in the High Speed Halo forum). But I also find it quite hard even to reach the correct landing spot. This could be due to PAL having lower power of drift, and possibly also due to MC's run-off speed being less than you can get with NTSC (MC runs several percent faster on NTSC I think).
As for NTSC, I get the impression (from what other people have said) that the fall to the ledge would be fairly easy after a bit of practice, though it probably takes some finesse to curve around the bulging cliff correctly. Avery Pawelek did some testing for me and reported a success rate of around 32%, while slYnki reports an estimated success rate of around 95%.
In a High Speed Halo thread 10l23r seemed to feel that the fall to the ledge is harder on PC than NTSC, and offered a detailed description of how he does it on PC. See here, and his slight correction here.
History and links
This method was first shown by xbill (using NTSC I believe). He advertised a video of it in the HBO forum on April 25th 2002, and added some explanation of his technique on May 4th (see also some later advice). The movie can currently viewed in the Tips and Tricks section at HBO. In his movie xbill's shield doesn't even flicker when he hits the ledge. He then heads along towards the Marines to get down off the ledge. He does so at a relatively dangerous place (it's not as friendly as it looks), taking his health down to only one health bar, but this was pioneering stuff so I don't think we'll hold that against him! His descent was enthusiastically greeted, and in HBO's Tips and Tricks section the comment partly reads "…and now xbill shows that it can be done, elegantly, in single-player mode, with no explosives or fancy moves. Just do the jumps correctly".
It seems that the method also got an earlier mention barely a week after the game was released. In a November 23rd 2001 'Battleground: Halo' forum thread which is now archived on the Wayback Machine, CanadianMonk3y said "jump to a small overhang on the left, and from there slowly hug the wall, and fall down. If done right, you can skip a large portion of the level, and for some reason, no enemies spawn!" In reply to a request for clarification he added "Well I was playing on legendary co-op and we sat there trying for like 15 minutes... He just randomly jumped off the first ledge, and hugged the wall and made it. Then I killed myself, and spawned by him, then I somehow made it onto a rock and we were down!" So, there wasn't much detail about the fall to the ledge (by the other player), but I think we can safely assume that it was essentially what xbill did later, albeit being done in co-op. However, I suspect that this fall soon got overlooked or forgotten because (a) it sounded like a pretty freaky event rather than a workably reliable method, and (b) CanadianMonk3y made a further post which readers may've taken to be an elaboration of the method but which was actually a quite separate co-op method based solely on respawning, not needing anyone to land on the snowy ledge. As such, I'd say xbill deserves the credit for bringing the method to proper notice, even if it did get this earlier mention.
The fall to the ledge can also be seen in Scurty's AotCR Bridge Fall Shortcut movie (NTSC). Rather usefully, he's looking down. Frankly that seems like a much better technique than seen in xbill's movie, as it enables you to better target where you're landing. It's also useful for viewers, to see where you're supposed to land! The landing place can likewise be well seen in cocopjojo's beautifully sharp AotCR Descent movie (PC). After getting to the ledge, Scurty heads left and drops down near the end of the battlefield, where the base of the cliff bulges out very nicely. Incidentally, if you're wondering why he's doing melee while going along the ledge, I asked him about that. Apparently it's just to stop from getting bored. Heh, these speedrunners eh?
You can also see Scurty using the method at 1:17 in his Legendary speed run. His method of getting to the bottom of the V is speedy indeed, comprising a drop straight down from the right rim of the deck (which I've been unable to do with PAL - which again I think is due to insufficient traction). After the fall to the ledge, he moves along up the battlefield then drops down to be cushioned by a large rock on the edge of the ice patch. The same techniques are seen in his PC demo AotCR Bridge Fall Shortcut (PC).
Update: slYnki has done a useful video tutorial "Halo 1 How To" - The AoTCR Bridge Fall - NTSC. I'll just mention though that when he starts jumping at 1:06 to delay the passage checkpoint, that was a touch later than I'd recommend. The checkpoint is triggered in the short section right after those Grunts as shown earlier in my guide, so to be safe you ought to start jumping by then (as seen in BCM40 and BCM41).
Another update: Chappified has performed a version of the method in his video AotCR - Shade and Tree assisted Bridge Drop on PC. Instead of stopping on the ledge, he slides straight off and manages to survive by bouncing off the top of a tree and then a Shade which he blasted down earlier. Tricky business! I gave my view on it in a reply to his his HBO forum post.