#RAW: the barbarian will not be disintegrated
In 5e, HP only goes down to 0. Once at zero, it doesn't change (though you still take damage you remain at 0 HP).
Disintegrate's must potent effect triggers only when it drops a creature's HP to 0:
If this damage reduces the target to 0 hit points, it is disintegrated.
However, it is impossible to reduce something to 0 HP when it is already there. This creates a bit of a loophole in the spell in a very subtle way which means a creature at 0 HP is immune to the disintegration effect of disintegrate.
Note that the Rage Beyond Death feature has no bearing on this loophole nor does it change the situation in any way beyond making the barbarian a more appealing target for disintegration since they are not unconscious. The loophole applies to all creatures at 0 HP equally.
#Rules as Intended: the barbarian is disintegrated
Jeremy Crawford, lead rules designer for 5e, confirms this loophole while also confirming that it was unintentional in this tweet:
The disintegrate spell has a loophole in it. RAI: The spell disintegrates you whether it deals damage to you when you're at 0 hit points or it reduces you to 0 hit points. RAW: You aren't disintegrated if the spell deals damage to you when you're already at 0 hit points.
Essentially what they intended to write was:
If this damage reduces the target to 0 hit points or deals damage to a target at 0 hit points, it is disintegrated.
So, your DM can decide which interpretation makes the most sense at their table.