30-year-old Weber debuted for the Chiefs in 2014 and established himself as the franchise's first choice halfback.
He was been stand-in captain for the Chiefs on several occasions during both the 2019 and 2020 Super Rugby seasons.
Cane first captained the Chiefs in 2016.
Since awarded the captaincy, Cane has captained alongside Aaron Cruden, Charlie Ngatai and Brodie Retallick.
Weber is honoured to undertake the role as a captain alongside Cane.
"I'm incredibly honoured to be asked to co-captain the Chiefs this year. A team that means the world to me so I don't take this responsibility for granted and I look forward to leading this exciting group of players with Sam this year.
"It's particularly special for me to co-captain alongside Sam, a man I have a huge amount of respect for in the way he plays but especially in the way he leads, so I can't wait to get stuck into the season with him and the rest of the lads."
Chiefs centurion Cane is looking forward to working alongside Weber in leading the squad.
"Webby has a massive amount of respect from the group, he consistently leads on & off the field for us, so the co-captaincy role is a natural fit for him and I am really looking forward to working along-side him in leading this exciting Gallagher Chiefs squad."
Head coach Clayton McMillan said Weber was a deserving recipient of the co-captaincy.
"Brad is an experienced player and highly respected figure in the Chiefs team. He has previously stepped into the captaincy role in the absence of Sam and/or Brodie over the past two seasons and I have every confidence in his ability to lead the team again alongside Sam. It's a huge honour and privilege to captain the Chiefs, and one Brad is deserving of."
The Chiefs first match of the Super Rugby Season is against the Highlanders on March 5 in Hamilton.