Combine tapioca, milk, and salt in a medium size saucepan on medium-high heat. Stir frequently until boiling. Simmer 5 minutes, uncovered at the lowest possible heat, adding sugar gradually.
Beat eggs in a separate bowl. Mix in some of the hot tapioca very slowly to temper the eggs. Return eggs to the pan with the tapioca. Slowly bring mixture barely to a boil stirring constantly. Reduce heat and stir several minutes until the pudding thickens. Cool 15 minutes. Add vanilla. Serve warm or chilled.