You can also use a mix of soy milk and other things to make delicious dairy free ice cream. I have had great success using a mixture of almond milk, soy milk powder and oil to replace the fat missing from the whole milk and cream and keep the ice cream creamy. There are are bunch of recipes on my site if you are interested. www.couldntbeparve.com
Kayt, if you're looking for a milk alternative, maybe consider coconut milk. I've seen recipes for coconut milk ice cream before (there are a few on the site), and the times that I've tried it, I've loved it. It will significantly affect the flavor, however.
It won't end up in disaster, but the fact there is no butterfat will cause the ice cream to crystallize and become rock-hard very quickly. In order to have the consistency you are looking for, you will have to serve/eat it all quite quickly.
Soy milk has a strong flavor (often sweetened) and different consistency. It's worth experimenting but I wouldn't expect them to come out the same way.
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It does not keep well.