#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.