You have no idea how annoyed I was reading through this thread. I’d come up with an idea and three posts further down there it would be.
But one that I didn’t see, hope I didn’t miss it while I was skimming parts.
Nowhere is it written that a locket has to hold a picture. Find one with a cavity large enough for the ring. Bonus is that you can probably still put a picture in it.
Good idea–I’ll check next time I’m at one.
Great idea, Projammer. Here’s a follow-up possibility: Add a small chain inside the locket, so that when the locket opens, the ring doesn’t automatically fall into the nearest non-recoverable spot. (If I had metalwork skills, I’d hack together a prototype.)
Yeah see I don’t like the locket idea because it just seems like it’s not secure enough. Attaching the ring inside the locket with a chain is starting to sound like an even bigger pain in the patootie than just unhooking the necklace and stringing the ring onto it in the first place, which is what I’m trying to avoid having to do. Also, I don’t want to wear a piece of jewelry that big as a daily “most of the time” kind of thing. I want something smaller and inobtrusive.
This looks like a more attractive version of the karabiner - it would be reasonably attractive if it came around to the front, would be easy to slip a ring onto and off, and very secure since I assume like similar smaller necklace clasps it’s spring loaded.
Here’s another which is sold by itself, to be added to any necklace.
Too late for edit window:
Not as sure how these work, but seem to be secure, come in a few designs, and could look kind of cool on a necklace.
This one has a clasp protector too.
You line up the cuts in the rings and slip them apart with a little effort.
OpalCat: While not completely mutually exclusive, super easy, super secure, and stylish is not going to be an easy mix. You’re most likely going to have to decide on a compromise solution that incorporates acceptable levels of each.
That, or just keep it in a cloth or leather bag in your pocket when not wearing it.
How about a wearing a long necklace, doubled over? To attach the ring, you’d just need to slip one loop of the necklace over your head, run it through the ring, and then slip it back over your head.
Kind of a long shot considering your criteria, but maybe a jewelry style, badge or ID lanyard with a snap hook would work as an alternative.
West Coast Lanyards might give you some ideas as well (pluss you can have customprinted/designed lanyards made there as well).
Here’s a cornucopia of different types of hooks.
For a simple, reliable, clasp …a kind of a classic, and sort of artsy, post modern look (at least to me) -I’ve always been a fan of simple Beaded Chains, like one finds on an old closet, attic, or garage, pull chain light bulb. I’m not talking about necessarily using it as the necklace, but the simple conjoiner to the ring and your choice of necklace. I admit that it would’nt be very sleek, just simple. I have no idea what that type of clasp or bead connector is called.
You might just want what is called a gate ring.
This looks like exactly the perfect thing.
Projammer: as I mentioned, I’m a vegetarian and am morally opposed to leather, but that aside, a lot of my clothes don’t have pockets, plus you can forget to take something out of your pocket and then it goes though the wash, gets lost, etc.
I’m pretty decidedly not looking for alternatives to taking the ring on/off of a short chain around my neck, I’m looking for the best way to put the ring on/off of a short chain around my neck.
Those gate ring things seem like the ideal solution.
I’m glad I could be of some help. I just thought of lanyard attache’ as a search start, as I have braided lanyards, beaded necklaces, and made some primitive jewelry as art projects at a couple of points in my life.
It’s funny though, because now I kinda feel like John Candy as Del Griffith, the Traveling Shower Curtain Ring Salesman in Trains, Planes, and Automobiles. 
Congratulations on your engagement, may you live and love out of bounds… happily, forever!
If those don’t work I have some very special Czechoslovakian Ivory Gate Rings that were originally crafted for the Grand Wizard of China back in the 4th Century. Now, they of course aren’t the originals, but they are replicas, very good replicas too. They are selling for five dollars a pair, two for seven dollars…
Have you tried these?:
http://www.firemountaingems.com/details.asp?PN=H207947JW
http://www.firemountaingems.com/details.asp?PN=H207948JW
http://www.firemountaingems.com/details.asp?PN=H207946JW
You can put the ring inside the purse and you don’t have to worry about it never looking stylish.
Assuming you end up with some sort of spring clasp/lobster claw instead of a toggle bar, you could also perhaps find or make one or two small charms to hang on the clasp as well. That way, regardless of how utilitarian the clasp itself looks, you still have some sort of pretty pendant-y things on the necklace at all times, even when wearing your ring on your hand. So it’ll always look like a necklace, rather than a pendant-in-waiting.
Bonus points if you can find/make charms that are in some way symbolic of your marriage. 
If you don’t decide on the gate ring you might want to check this page.
Half way down are lobster claw clasps that are shaped like hearts, cats & elephants.
Give these a look - charm holders. Ridiculously expensive if you want gold, but pretty enough to wear all the time.
With a sufficiently flexible necklace, you can just pull a loop of the necklace through the ring and pass this loop over the part of the necklace that is on the other side of the ring. As long as the necklace is around your neck, the ring can’t come off. Once you take off the necklace, the ring comes off easily.
I’ve no idea how this ring pendant works, but I saw this advertised in the What on Earth catalog and thought of you, Opal.
It’s tough to tell how the hingeing is supposed to work - it rather looks like you need to take off the necklace to put the ring on, which kind of defeats your OP - but it might be worth an inquiry.
If you are talking about a cow hitch, it’s been discussed already in half a dozen posts at least. If you’re talking about doubling over a longer necklace, it’s been mentioned several times as well.
I pretty specifically don’t want to have to take off the necklace to do this, don’t want to deal with a knot, and I don’t want a necklace longer than my chin.
I know a few people have said I’m being too picky and will never find something that meets my needs, but let me point out that I’ve already said that several things mentioned in this thread fit my needs perfectly (thanks guys!). I haven’t bought one yet, but I plan to, once I narrow it down.
Yeah that is puzzling. Now I’m going to be stuck wondering how it works! Thanks a lot! 
The ring is obscuring the hinge in that photograph. It swivels and gates at the antapex of the heart. The apex and “vulva” of the heart is the thread. It must be removed from the chain to heave the split.