We report the first experimental demonstration of one-dimensional pola
rization gradient cooling of bound ions. The trapped particles are onl
y weakly bound along the laser cooling axis. Nevertheless, the particl
e motion has to be treated quantum mechanically. The minimum kinetic e
nergy is found when the depth of the optical potential approaches a qu
antum of the vibrational energy of the trapped ions. Cooling to low vi
brational states of the trap has been observed. We have also produced
the lowest one-dimensional temperature for an ion crystal.