The 27-year-old has agreed to a three-year deal at Portman Road and will leave relegated City as a free agent.
"I loved my time at Bristol, they gave me a chance in the game and I'll always be grateful for that," Cole told Ipswich's official website.
"It's the right time for me to move on though. I need a fresh challenge and when I knew Ipswich wanted me, I didn't have to think too hard about the decision.
"I had a few options, but with Mick McCarthy and Terry Connor in charge at Ipswich, that shows the intent of the club to move forward and I can't wait to get going."
Skuse progressed through the academy ranks at Ashton Gate and went on to make more than 300 appearances for his hometown club after making his debut in 2005.