Step 1In a medium, nonreactive saucepan, stir together the sugar, one-half cup water, the orange juice and the rum. Gently crush the sprig of mint in your hand to release the oils, and place the mint in the pan. Bring the mixture to a simmer over medium heat, about 5 minutes, and continue to simmer for 2 to 3 minutes to marry the flavors. Remove from heat.
Step 2While the liquid is coming to a simmer, whisk together the cornstarch and remaining one-half cup water to form a slurry. Stir the slurry into the pan, then heat the pan over medium-low heat and cook until the liquid thickens (it should just begin to bubble), 3 to 4 minutes, stirring constantly. Remove from heat and discard the mint sprig.
Step 3Arrange the strawberries in the prepared crust. Pour the thickened filling over the berries, then refrigerate the pie 3 to 4 hours to set the filling.