Exclusive

Miley Cyrus and Liam Hemsworth are officially single in the eyes of the law ... because we've learned their divorce is now final.

According to new legal docs obtained by TMZ, a judge has already put their John Hancock on the former couple's divorce settlement.

TMZ broke the story ... Miley and Liam worked out all the details of their divorce back in December, and their lawyers -- disso queen Laura Wasser repping Liam and Judith R. Forman repping Miley -- quietly worked through stumbling blocks to submit the paperwork to the judge.

It's a pretty quick divorce ... Miley and Liam have no kids, she's keeping their pets and our sources say there's a prenup, which makes dividing property a helluva lot easier.

Liam's the one who filed to divorce Miley back in August, citing irreconcilable differences ... and now they are both single.

Miley has been very public about her new relationship with singer Cody Simpson, and Liam has been a little more on the DL with his.