The deal runs from the start of the 2014/15 season and means he’ll be at Scotstoun until at least May 2016.
Finn is a former Scotland U20 stand-off who was attached to Glasgow Warriors on an EDP contract.
The 21-year-old played for Lincoln University in Canterbury, New Zealand over the summer, after becoming the latest recipient of the Macphail Scholarship. Following a four-month spell in New Zealand, Finn was voted the Club Player of the Year.
Finn played his club rugby with Falkirk but moved to Ayr in season 2012/2013.
Before turning pro, Finn worked as a stonemason.