Stevia is SO much sweeter than sugar, you have to be very careful when you are subbing it in a recipe that calls for sugar. Additionally, without sugar, your ice is likely to freeze like a brick. The sugar keeps the water from freeing solid. Stevia won't do this.
You could also add a little bit of lemoncello or citrus vodka if you want the ice to be softer. It should help do the job the sugar normally does, provided everyone who will be consuming it is of age.
You could, but it's hard to say how it might affect your end product. Sugar in frozen treats not only adds sweetness, but it also alters how the water freezes. Sugar helps bind water, so smaller crystals form in the product as it freezes. You can overcome this to some extent by interrupting to process to break it up so it has a finer final texture. If you do a web search, there will be recipes for frozen lemonade and popsicles using stevia as a sweetener, and that might give you some place to start experimenting.
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Additionally, without sugar, your ice is likely to freeze like a brick. The sugar keeps the water from freeing solid. Stevia won't do this.