Heat, in a medium-size heavy saucepan, over medium heat 3 cups of milk, cinnamon stick, lime zest, sugar, and salt. Bring to a boil, uncovered. When it starts to boil reduce the heat to low, add rice and cook for 10 - 15 more minutes or until milk is almost evaporated. Continue adding milk, stirring constantly, 1/2 cup at the time. Wait to add more milk until the liquid has been almost completely absorbed by the rice. With the last ½ cup of milk also add the condensed milk; cook for 2 -3 minutes, or until it thickens to desired consistency. Be aware that as the rice cools it thickens. Turn off the heat.

Add butter and stir to incorporate.

Let cool uncovered, until it reaches the desired temperature. You can serve it hot, warm, or cold.

Serve hot, warm, or cold sprinkled with ground cinnamon to taste.