Saving throws are always made against a DC, and in the case of casting spells there is a formula to work out the required DC. From the Basic Rules on spellcasting:

Saving Throws

Many spells specify that a target can make a saving throw to avoid some or all of a spell's effects. The spell specifies the ability that the target uses for the save and what happens on a success or failure.

The DC to resist one of your spells equals 8 + your spellcasting ability modifier + your proficiency bonus + any special modifiers.

And as the Bard class description says:

Spellcasting Ability

Charisma is your spellcasting ability for your bard spells. Your magic comes from the heart and soul you pour into the performance of your music or oration. You use your Charisma whenever a spell refers to your spellcasting ability. In addition, you use your Charisma modifier when setting the saving throw DC for a bard spell you cast and when making an attack roll with one.

Spell save DC = 8 + your proficiency bonus + your Charisma modifier

So your bard spells have save DC based on your charisma modifier and proficiency bonus as described above. In general, whenever a class or racial feature allows you to cast spells somehow, its description will clarify what the relevant spellcasting ability modifier is, if any, so that you can work out the saving throw DC.

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.