Combine currants and Port in a small bowl and set aside to plump.
Wash and drain the rice. Place in a large saucepan with plenty of water to cover. Stir, and bring to a boil over medium heat. Boil for 5 minutes, to blanch the rice. Drain rice in a colander, rinse with warm water, drain again and set aside. While rice is cooking, place 3 cups milk in a medium heavy bottomed saucepan, add cinnamon sticks, lemon zest, and salt. Bring to a simmer over medium heat. Turn off heat and let infuse until rice is ready.
Place rice in another medium heavy bottomed saucepan, and ladle in 1 cup warm milk. Stir well with a wooden spoon, then turn on the lowest possible heat and cook gently, stirring occasionally, until milk is absorbed. Ladle in 1/2 cup warm milk, stir, and cook until absorbed. Continue in this manner until you have used up all the milk (remove and discard the cinnamon sticks). With the last 1/2 cup of milk, stir in also the sugar, butter, ground cinnamon, and the plumped currants in Port. When the last addition of milk has been absorbed, turn off the heat and evaluate the consistency against your personal rice pudding preferences, bearing in mind it will be thicker when chilled. Add 1/4 cup to 1 cup additional milk (cold is fine) as desired. Turn pudding out into a medium serving dish or divide between six individual serving dishes and decorate top with ground cinnamon.
Portuguese rice pudding is cooked entirely on top of the stove in a risotto like manner. It traditionally is thickened and enriched further with egg yolks, but has a voluptuous texture even without them, and is lower in fat. Stir the rice frequently but NOT constantly because it can become too gummy. The classic Portuguese seasonings are lemon and cinnamon, but not vanilla. Use ground cinnamon to decorate the top; easiest way is sifting through a doily.
Port, or Porto, comes from Portugal and is considered one of the great dessert wines of the world. It is a fortified wine, meaning that brandy is added. Most Port is made from red grapes and is served after the meal. Ruby Port is full, fruity and young; tawny ports have been aged in casks and are mellower. One can also find white Port, which is made from white grapes, and is drunk before the meal.