With a trip to the toy store and an hour of snapping and unsnapping LEGOs, I came up with a little gearbox connecting a motor to a stirring rod. The gears give this light-weight motor the mechanical advantage that it needs to churn thickening ice cream. For each turn of the rod, the motor spins 15 times.
The assembly — shown with some spare pieces attached as stirring implements — fits snugly into a large empty yogurt container with notches cut in the sides.
Ice cream is milk : cream : sugar in the proportion 2 : 2 : 1 with a pinch of salt and flavoring. Just mix them and agitate the mixture while it freezes. (Other, more complicated recipes involve heating the milk and using eggs too. No need for more complication here.)
I let it chill in peace for about an hour. At 7:30, when I could see ice crystals starting to form, I switched on the motor, headed out the door for the evening, and hoped for the best. Around midnight, I came home to a delicious success!
Thanks to Jeff Potter for sharing the idea of LEGO ice cream makers in his book. With the help of the enthusiastic people at Shananigans Toy Shop, I chose LEGO Power Functions (#8293) for a motor and LEGO Crazy Action Contraptions (Klutz Press) for a starter set of bricks.