Weston has agreed a two-year deal with Mark Stimson's side that will keep him at the club until the end of the 2009-10 season.
He made only three starts during his season at Elland Road and had a period on trial with the Scottish Premier League side Hibernian last month.
Weston started his career as a trainee at Millwall before moving to Swindon Town two years ago and making 31 starts for the Robins.
He was loaned out to Scunthorpe United for the final two months of last season and featured in seven games for Nigel Adkins' side.