susan g, I've made sweetened yogurt with both stevia and xylitol, but on your recommendation I'll go back to xylitol for my frozen yogurt. Why do you think xylitol is a better sweetener than stevia? Honey could be interesting...

Another possible factor: the sweetener. Stevia has no bulk and none of the chemical properties of sugar which might give you better texture. How about xylitol, a 'real' but better sweetener? or honey?

Hmmm... AntoniaJames, I think you're right; insulating could be the way to go. I could also check the temp with my Thermapen (instant-read thermometer) and learn when to remove it for consumption, as further storage would continue to lower the temp... Thanks ever so much...!

Thanks so much for this follow up and the info on yogurt making! Have you considered insulating the container in the freezer (before the frozen yogurt becomes rock hard), so that the yogurt is not held at such a low temperature? I'd experiment with that, as the temperature is probably the largest contributing factor, and probably also the easiest to control. ;o)

Thanks, everybody...! I'm going to experiment: because I take no alcohol, vodka and other liquors are out; I'm going to strain my yogurt and add a bit of gelatin; I avoid sugar but use stevia. And I've been making low-fat but I'll try whole milk and see if that helps.

AntoniaJames, I usually avoid promoting products, but I'll make an exception here. My yogurt maker is a New Zealand product called Easiyo, bought from Amazon.com for around $25. It's basically a thermos; you prepare your yogurt-to-be, boil a pot of water which goes into the thermos and you're set. They want to sell you their expensive packets of dry prep, so I buy a few packets then make generation after generation of yogurt (adding 1/2 cup of dry milk for thickness) from each packet, freezing a couple of tablespoons from each batch to start the next. I cannot recommend the Easiyo too highly. Couldn't be easier.

By the way, how do you make your own yogurt? I've been trying to make yogurt, but haven't been thrilled with the results, so I am interested in knowing what you do to make it turn out so well. Thanks!!

Water is probably what's making it freeze rock solid. Try straining your yogurt in the refrigerator for at least 6 six hours through a large fine-mesh sieve placed over a bowl, or use a colander line with cheesecloth or paper coffee filters.

Are you making whole-milk yogurt? The additional fat helps with the texture.

You could add one tablespoon of vodka per two cups of yogurt; you could add a little extra sugar or honey--you should be using between 1/2 to 2/3 cup of sugar or honey for each two cups of yogurt; or, if none of these suggestions makes the texture creamy, put sticks in it and have yogurt pops.

P.S. I meant lower the freezing temperature. Oops.

While I agree that alcohol will raise the freezing temperature and therefore make it less hard, I think you also have to constantly stir it as an ice cream maker would do, or as you do when making a granita. (Let it freeze slightly and then break it up with a fork. Repeat until it's frozen. Then let it sit out for about 10 minutes before serving.)
Butter and sugar also prevent it from freezing too hard so you could make some buttered caramel and swirl that through as you're breaking up the yogurt.

A bit of gelatin also gives it a great mouth feel, by the way. You need to take the yogurt out periodically and stir it around. Also, have you considered getting a Donvier frozen dessert maker? I have one that I bought in the last century, but I was curious and looked them up on Amazon, to find that they are still available. They are not too expensive, and don't take up much space. If you only want to make a pint at a time, they're great. I like TiggyBee's idea of adding vodka. You could add another liquor like rum or whiskey, if the flavor were compatible/interesting. ;o)

I learned the trick of putting some vodka in it, since alcohol doesn't freeze from "The Perfect Scoop" by David Lebovitz.