Exhaustion is a special condition which has multiple levels. From the PHB p. 291, all emphasis mine:
Level Effect
⋮
6 Death
Exhaustion is measured in six levels.
A creature suffers the effect of its current level of exhaustion as well as all lower levels.
Death ward is a 4th level spell which, amongst other things, has the following effect (PHB p. 230):
If the spell is still in effect when the target is subjected to an effect that would kill it instantaneously without dealing damage, that effect is instead negated against the target, and the spell ends.
This means that since the effect of the sixth level of exhaustion is instantaneous death—in that it happens at the instant the sixth level is gained—then the quoted clause of death ward triggers, upon which the spell ends. However, since the level of exhaustion is not removed by the spell—only its effect negated—it is unclear how to resolve the situation which results: does the creature 'suffer the effect' of the six levels of exhaustion, dying (again), or is there any basis to another interpretation?
This begs the question: what happens when a creature under the effect of death ward suffers the sixth level of exhaustion?