The 6ft 7in centre-half, 21, spent most of last season on loan at Yeovil, whom he helped win promotion to the Championship.
Burn told Birmingham's official club website: "I'm really happy to have joined and I'm just looking to get back to pre-season training now.
"The history of this club was something that convinced me to join.
"Lee Clark, having played for Newcastle, is also somebody that I knew a lot about too and there's a lot of good players here. Now it's up to me to go and prove myself."
Burn, who scored Yeovil's winner in their League One play-off final triumph over Brentford, started his career at Darlington in 2009 and joined Fulham at the end of the 2010/11 season. He recently signed a new two-year deal at Craven Cottage.
The Blues signed fellow defender Kyle Bartley on loan from Swansea on Monday and have also completed the permanent signing of striker Matt Green from Mansfield.
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