Hunter and Kathleen Biden have settled up their divorce just 4 months after she filed, and 6 weeks after his relationship with his brother's widow went public.

According to court docs ... Joe Biden's son and his ex reached an agreement on "all questions relating to custody, support, property rights, and all other rights." Specific terms of the settlement were not disclosed.

Kathleen had sought sole physical and legal custody of their minor child, while Hunter wanted joint custody -- a big gap they've clearly bridged. Both sides made accusations of cheating ... before agreeing to quit the mudslinging in public.

As of Friday ... they're both officially single. Hunter's now dating Hallie Biden, who was married to his brother, Beau.