Fill double-boiler pan partially with water so that water will not touch the bottom of the double-boiler insert bowl. Cover pan (without insert) and place over high heat to bring to boil.
In double-boiler bowl, combine eggs and milk. Stir in tapioca and sugar.
Place bowl over boiling water and cook uncovered without stirring for 7 minutes.
Stir and cook for another 5 minutes.
Remove bowl from heat. The tapioca thickens as it cools. Fold in vanilla.
Leave in bowl or transfer to serving dish(es). Chill in refrigerator.
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