Both Uses of

immortal
in
The Count of Monte Cristo

Do what I tell you, and immortality awaits you!†
Chpt 79-80 *

immortality = eternal life (to live forever)

Suppose that the Supreme Being, after having created the world and fertilized chaos, had paused in the work to spare an angel the tears that might one day flow for mortal sins from her immortal eyes; suppose that when everything was in readiness and the moment had come for God to look upon his work and see that it was good—suppose he had snuffed out the sun and tossed the world back into eternal night—then—even then, Mercedes, you could not imagine what I lose in sacrificing my life at this moment.†
Chpt 89-90

Definitions:

(1) (immortal)
living or existing forever

or:

someone famous throughout history

or:

someone who will never die -- such as a mythological god
(2) (meaning too rare to warrant focus)
meaning too rare to warrant focus:

More rarely, "The Immortals" denotes a military corps of the Persian Empire. The Immortals were so-named because each time a member of the 10,000 man corps was killed or seriously wounded, he was replaced by another man. They are best remembered in western culture for their role in defeating the badly out-numbered Spartans at the Battle of Thermopylae.