Baun secured a one-year deal with the Belfast Giants (EIHL) on Wednesday.
The Canadiens traded Baun and Tomas Plekanec to the Maple Leafs on Sunday, with Rinat Valiev and Kerby Rychel heading the other way, TSN reports.
Baun was acquired by Montreal for Andreas Martinsen on Wednesday.
The Blackhawks waived Baun on Friday, Renaud Lavoie of TVA Sports reports.
Baun was recalled from AHL Rockford on Tuesday.
Baun was assigned to AHL Rockford on Saturday.
Baun signed a two-year contract extension with the Blackhawks on Thursday, Tracey Myers of CSN Chicago reports.
Baun was assigned to AHL Rockford by the Blackhawks on Friday.
Baun tallied two goals and an assist in Friday's exhibition win over the Canadiens.