Combine first 8 ingredients; stir well. Beat butter in a large bowl at medium speed of an electric mixer until creamy; gradually add sugar, beating well. Add egg, beating until blended. Stir in rum. Gradually add flour mixture, beating just until blended after each addition. Cover and chill dough 45 minutes.
Shape dough into a 13" log on wax paper; wrap and freeze log 1 hour or until firm.
Cut frozen log into 1/4" slices, and place on lightly greased cookie sheets. Bake at 350° for 12 minutes. Remove to wire racks to cool completely.