Mix the rice and warm water together in a bowl, and let stand for 1/2 hour. Reserving the water, drain, and place the rice in the bowl of a food processor. Add the cinnamon and process until a paste forms. Return the rice to the water and let stand at least 2 hours, stirring occasionally as the water turns milky white.

Strain the rice through a fine sieve into a bowl or pitcher. Stir in the milk, condensed milk, vanilla, and rum, if desired, until evenly blended. Refrigerate at least 2 hours.

To serve, divide the ice cubes between four glasses, and pour the chilled horchata over the ice.