Amateur
Weissbach played at the junior level in North America for the Tri-City Storm of the United States Hockey League (USHL), and was named to the Third All-Star Team of the 2016–17 USHL season.
Weissbach committed and played for the University of Wisconsin. In his freshman season, he was named to the Big Ten Conference's All-Rookie Team. In his senior season, he helped Wisconsin win the Big Ten Conference and qualify for the NCAA national tournament. He was third in the nation with 41 points and second with 29 assists in 31 games. He was named to the Big Ten's Second All-Star Team, became a second-team All-American and received the Big Ten Medal of Honor. In his college career, Weissbach also received Academic All-Big Ten honors three times.
Professional
Weissbach was drafted in the seventh round, 192nd overall by the Buffalo Sabres in the 2017 NHL entry draft. Following the completion of his four-year collegiate career, the Sabres signed Weissbach to a two-year entry-level contract on 6 April 2021. At the conclusion of his entry-level contract, he signed a one-year, two-way extension with the Sabres.
Having played exclusively with Buffalo's AHL affiliate, the Rochester Americans, Weissbach left the organization as a free agent and returned to his original Swedish club, Frölunda HC of the SHL, on a two-year contract on 20 July 2024. He had surgery on his shoulder in 2025 and missed a part of the season.
In 2026, Weissbach signed a one year contract with SKA Saint Petersburg, competing in the Russian Kontinental Hockey League (KHL). He was the first Swedish player to switch from SHL to KHL since Russia's invasion of Ukraine.
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