In a large saucepan, combine the milk, 1 cup sugar, the salt and lime zest and bring to a boil. Reduce the heat to medium-low and simmer for 15 minutes, stirring occasionally. Strain and let cool.
Preheat the oven to 350 degrees . In a medium skillet, melt the remaining 1 cup sugar over medium heat. Cook until light golden, about 5 minutes. Pour the caramel into a 9-by-3-inch round cake pan and swirl it around the bottom to coat.
In a medium bowl, beat the eggs until just combined, then stir in the cooled milk mixture; strain. Pour the custard over the caramel and set the pan in a large roasting pan. Add 1 inch hot water to the roasting pan and bake the custard until set, about 1-1/4 hours. Remove the custard from the roasting pan and let cool completely on a wire rack; cover and refrigerate until chilled, 6 hours or overnight. To serve, run a knife around the edge of the pan and invert onto a serving plate.