Weber got his start in the sport at an after-school job in a bowling alley, where the proprietor allowed him and other pinsetters to bowl without charge during off-hours. It was the beginning of one of the most successful bowling careers in history.

After years of bowling for small sums in after-midnight matches, while holding jobs ranging from street photographer to postal clerk, Weber was invited in 1955 to join one of the first professional bowling teams in the United States, the Budweisers, located in St. Louis, Mo. In his first tournament appearance with the team, which included future legends Don Carter (Carter, Don), Ray Bluth, Tom Hennessey, and Pat Patterson, Weber's share of the prize was $200, equal to what he had earned in four weeks at the post office.

The Budweisers dominated the fast-growing sport in the late 1950s, and Weber soon became one of the most frequent winners in individual tournaments held by the PBA after its formation in 1958. He won four national individual match game championships and paired with Bluth to capture the national doubles title four times. Weber continued his string of victories even after being elected to the PBA Hall of Fame in 1975. He became the first bowler to win PBA titles in six consecutive decades when he won a PBA title in 2002.

The youngest of Weber's four children, Pete Weber, followed his father's footsteps into the PBA Hall of Fame in 1998.