9 . Slowly lower the temperature to -173 centigrade or lower, as you wish.

If I'm reading the chart correctly, the additional cooling would send the ice III through the zone marked as ice II and then... wait for it... into the zone of ice nine!!!

If the secret of eternal life involves the non-fictitious version of ice IX... I mean... that seems like "the author" would be clubbing us over the head with the implication that we're living in a post-modern novel :-P

On a less metaphysical note, it seems like there is a technical question about whether additional cooling might cause problems due to transitions between different kinds of ice? From Le Wik on the real ice IX (not the fictional ice-nine):

Ice IX is a form of solid water stable at temperatures below 140 K and pressures between 200 and 400 MPa. It has a tetragonal crystal lattice and a density of 1.16 g/cm³, 26% higher than ordinary ice. It is formed by cooling ice III from 208 K to 165 K (rapidly—to avoid forming ice II). Its structure is identical to ice III other than being proton-ordered.

It looks like if you were in the ice IX zone, and then heated up from LN2 temperatures, you would necessarily go through ice II on the way to liquid water (see this awesome site):

Ice-nine (ice IX) is the low-temperature equilibrium, slightly denser, structure of ice-three (Space group P41212, cell dimensions 6.692 Å (a) and 6.715 Å (c) at 165 K and 280 MPa [385]). It is metastable in the ice-two phase space and converts to ice-two, rather than back to ice-three, on warming. The change from proton disordered is a partial process starting within ice-three that is only completed at lower temperatures, but with a first order transition near 126 K[1087]. The hydrogen bonding is mostly proton-ordered as ice-three undergoes a proton disorder-order transition to ice-nine when rapidly cooled in liquid nitrogen (77 K, so avoiding ice-two formation, see Phase Diagram); ice-three and ice-nine having identical structures apart from the proton ordering [389].

From what I can tell, if you start at ice III and cool things way down from there, you'll have to spend some time in the ice II zone, at the very least while being re-heated up from ice IX and perhaps as the state to be kept in for very long term storage. Luckily, ice II appears to also have a density of ~1.16 g / cm^3, so it is also denser than normal water and presumably would also not pop cellular membranes due to expansion :-)