#RAW: the barbarian will be disintegrated

The rules now say this explictly per the 2018 PHB errata to close what was previously a loophole.

The target is disintegrated if this damage leaves it with 0 hit points.

Since, the barbarian would be at 0 HP after taking the disintegrate damage they would be disintegrated.

#The loophole was never intended

Jeremy Crawford, lead rules designer for 5e, confirms this loophole while also confirming that it was unintentional in this tweet (made before the 2018 errata):

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.

RAW: the barbarian will be disintegrated

The rules now say this explictly per the 2018 PHB errata to close what was previously a loophole.

The target is disintegrated if this damage leaves it with 0 hit points.

Since, the barbarian would be at 0 HP after taking the disintegrate damage they would be disintegrated.

The loophole was never intended

Jeremy Crawford, lead rules designer for 5e, confirms this loophole while also confirming that it was unintentional in this tweet (made before the 2018 errata):

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.

#RAW: the barbarian will not be disintegrated

The rules now say this explictly per the 2018 PHB errata to close what was previously a loophole.

The target is disintegrated if this damage leaves it with 0 hit points.

Since, the barbarian would be at 0 HP after taking the disintegrate damage they would be disintegrated.

#The loophole was never intended

Jeremy Crawford, lead rules designer for 5e, confirms this loophole while also confirming that it was unintentional in this tweet (made before the 2018 errata):

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.

#RAW: the barbarian will be disintegrated

The rules now say this explictly per the 2018 PHB errata to close what was previously a loophole.

The target is disintegrated if this damage leaves it with 0 hit points.

Since, the barbarian would be at 0 HP after taking the disintegrate damage they would be disintegrated.

#The loophole was never intended

Jeremy Crawford, lead rules designer for 5e, confirms this loophole while also confirming that it was unintentional in this tweet (made before the 2018 errata):

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.

#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 the target has 0 hit points after taking damage from this spell, it is disintegrated.

So, your DM can decide which interpretation makes the most sense at their table.

#RAW: the barbarian will not be disintegrated

The rules now say this explictly per the 2018 PHB errata to close what was previously a loophole.

The target is disintegrated if this damage leaves it with 0 hit points.

Since, the barbarian would be at 0 HP after taking the disintegrate damage they would be disintegrated.

#The loophole was never intended

Jeremy Crawford, lead rules designer for 5e, confirms this loophole while also confirming that it was unintentional in this tweet (made before the 2018 errata):

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.