Wiegert attended the University of Nebraska-Lincoln, where he played for the Nebraska Cornhuskers football team from 1991 to 1994. From his sophomore through senior seasons, he started 37 consecutive games at right tackle. In his 46-game Cornhuskers career, Wiegert gave up just one quarterback sack while earning first-team All-Big Eight Conference honors in 1992, 1993 and 1994. He earned second-team All-America honors in 1993, helping lead Nebraska to an undefeated regular season and an Orange Bowl appearance.
As a senior in 1994, he was the winner of the Outland Trophy, awarded to the nation's top interior lineman, and was recognized as a consensus first-team All-American. Wiegert blocked for a Cornhuskers offense that led the NCAA in rushing yards for the eleventh time, averaging 340 yards per game on the ground, and Nebraska won its first national football championship since 1971. He was also the UPI Lineman of the Year, the Touchdown Club of Columbus Offensive Lineman of the Year, and a finalist for the Lombardi Award.
The university retired his No. 72 jersey before the 1995 season.
The St. Louis Rams selected Wiegert in the second round (38th pick overall) of the 1995 NFL Draft, and he played with the Rams from 1995 to 1998. As a rookie in 1995, he played in five games and started two; he was inactive for the first seven games with a left ankle injury suffered in final preseason game. In 1996, he started all 16 regular season games at right tackle. The following year, under new coach Dick Vermiel, Wiegert moved to right guard and started 15 of 16 regular season games. He caught a one-yard pass to give the Rams a first down on game-winning drive that resulted in Jeff Wilkins' 25-yard field goal with four seconds remaining at Washington (11/30/1997). He also scored first-career TD after recovering Jerald Moore's fumble in the end zone at Carolina (12/20/1997). He was granted free agency on February 13, 1998, and re-signed with Rams on June 17, 1998. In 1998 he started 13 games at right guard for St. Louis. He missed three games with a knee injury. He was designated by Rams as transition player on February 12, 1999, and was re-signed by the Rams on March 24, 1999. He was released by Rams on April 28, 1999, after had become expendable since Rams had signed Adam Timmerman to play right guard.
Wiegert played four seasons with the Jacksonville Jaguars from 1999 to 2002. He signed as a free agent May 5, 1999, with the Jaguars in 1999. That season he played in 16 games with 12 starts at right guard for 14-2 Jaguars. In 2000 he started eight games at right tackle before suffering a torn anterior cruciate ligament in left knee. He had allowed only four sacks in eight games. He was placed on injured reserve on October 14, 2000. In 2001 he came back from injury to start all 16 games at right guard, the first time he had been able to do so since 1996. Wiegert played and started in seven games in 2002 before being placed on injured reserve on 11/6/2002. He made his first NFL start at left tackle versus Indianapolis and started four more games at left tackle before starting two games at right tackle. He was granted unconditional free agency on February 28, 2003.
He played for the Houston Texans from 2003 to 2006. He signed by Texans on March 1, 2003. In 2003 he played in 15 games, starting 14 at right guard. In 2005 he started 12 games (five at right guard, seven at right tackle). The following season, 2006, he played in and started nine games at right tackle. On February 28, 2007, the Texans released Wiegert.