Pour sugar into a heavy saucepan over medium-low heat and stir with a whisk until it starts to break down into very fine granules.

Add butter and stir until completely melted and bubbly (about 2 minutes). Stir in rum.

Slowly pour in heavy cream until it comes to a boil, stirring constantly. Remove from heat and allow to cool 5 minutes. Pour over bread pudding just before serving.