We investigate the final state of zero-temperature Ising ferromagnets which are endowed with single-spin flip Glauber dynamics. Surprisingly, the ground state is generally not reached for zero initial magnetization. In two dimensions, the system either reaches a frozen stripe state with probability approximately equal to 1/3 or the ground state with probability approximately equal to 2/3. In greater than two dimensions, the probability to reach the ground state or a frozen state rapidly vanishes as the system size increases and the system wanders forever in an iso-energy set of metastable states. An external magnetic field changes the situation drastically -- in two dimensions the favorable ground state is always reached, while in three dimensions the field must exceed a threshold value to reach the ground state. For small but non-zero temperature, relaxation to the final state first proceeds by the formation of very long-lived metastable states, similar to the zero-temperature case, before equilibrium is reached.

10 pages, 17 figures, 2-column revtex format