Heat water and apricots to boiling; reduce heat. Cover and simmer until tender,
about 20 minutes.
Place apricots and 1/2 cup cooking liquid in blender container;
cover and puree until uniform consistency.
Press puree through sieve.
Mix sugar, cornstarch and salt in saucepan; gradually stir in apricot puree and
remaining cooking liquid. Heat to boiling over medium heat, stirring constantly.
Boil and stir 1 minute. Pour into dessert dishes.
Serve with half-and-half or sweetened whipped cream if desired.