It is possible for the temperature of ice to be lower than 32 degrees Fahrenheit. At 32 F, water begins to turn to ice. However, if the environment surrounding the ice cube is colder than 32 F, the ice cube drops to that temperature.
In general, whenever the environment surrounding the ice cube is colder than the cube, the cube loses heat. For example, freezers are typically colder than 32 F, and the ice inside the freezers match their temperatures. In a more extreme case, if an ice cube is placed in liquid nitrogen, the cube's temperature drops to match that of the nitrogen, which is negative 320 F.