To prepare the pan, grease and flour a 10” bundt cake pan. Sprinkle the chopped walnuts in the prepared pan.
Combine the cake mix, pudding mix, rum, water and oil. Beat well.
Add eggs one at a time, beating between each addition.
Pour the batter into the prepared bundt pan.
Bake for 60 minutes.
While the cake is baking, prepare the glaze. In a saucepan over medium low heat, mix together the sugar, water and butter. Bring this to a low boil and simmer for 5 minutes, stirring constantly. Remove from heat and stir in the rum. (If you’d like to cook the rum, you can! That will remove a very slight amount of the alcohol.)
Prick the top of the cake with a fork and while the cake is still in the pan, pour the glaze over the cake. Give it at least 30 minutes to sink into the cake.
Turn the cake out onto a plate to cool completely.
This cake is best when it has time to sit. Store it in an airtight container and serve it the next day.