I finally found cardboard ice cream containers! These quart containers  are made by Sweet Bliss, and come 25 to a pack. I’ve used tons of them already, and they’re perfect. They also have the same thing in pint size . You can also buy them directly from the Sweet Bliss web site .
If you’re outside the USA and having trouble ordering the Sweet Bliss containers, try ordering the 32oz “Paper Food Containers” from Ace Mart . I’ve used them as well, and they’re just as good. Order the clear plastic lids from them separately. They’re in the USA but they seem to ship overseas.
In the UK, try ordering the 16oz and 32oz “Paper Soup Containers” from Cater For You . Order the paper or clear plastic lids from them separately.
Let me know if you hear of another place that sells containers like these in small quantities.