Yes, the universe will die one day, but we do not exactly know how that will happen. There are two possibilities. In the first case, the expansion that started with the birth of the universe will continue, but will become slower and slower. At some point of time, the last star will die and all matter will disappear in the black hole. The universe will be nearly empty and cold with a weak radiation. In the second case, just the opposite will happen. The universe will stop expanding at some point of time. The attraction forces of all the objects will then cause the universe to collapse on itself. The universe will be infinitely hot and small.