Grate the zest of each lemon and mix it with the sugar in a large bowl. Set aside for 10 minutes. Preheat the oven to 320°F (160°C).
Melt the butter in a bowl and let it cool.
Add the eggs one at a time to the sugar-lemon mixture, mixing between each addition.
Add the cream, rum, and salt, and mix.
In a bowl, mix the flour and baking powder, and add the mixture to the previous mixture in 3 additions, mixing between each addition.
Add the melted butter.
Pour the cake batter into a large cake pan and bake for 70 to 80 minutes in an oven preheated to 320°F (160°C).
Insert a knife into the middle of the cake to verify that it is cooked through; the knife should come out clean.
To prepare the syrup, heat the sugar and water in a saucepan. Remove from the heat as soon as it starts to boil. Remove from the heat and add the lemon juice.
When you take the cake out of the oven, pour the syrup, warm or cold, over the cake (still in the pan). Let cool thoroughly for at least 1 hour before removing from the pan.
Good to know: these proportions make one large cake and two small cakes.
