Rather than play in Canadian juniors, Conn opted to keep his American university eligibility alive and he eventually settled on the University of Alaska-Anchorage. In his three seasons there from 1988 to 1991, the big winger established himself as one of the best players in the school's history.
Conn went undrafted by the NHL but was signed as a free agent by the Chicago Blackhawks upon graduation in 1991. Although he would get a brief 2-game look in the NHL in his rookie professional season, Conn spent the next 3 and 1/2 seasons with the Hawks farm team in Indianapolis where he was one of the more popular players in that city's hockey history. He was used primarily in a checking role in which he excelled at that level.
In 1995 the Hawks traded Conn to the New Jersey Devils organization in exchange for aggressive defenseman Dean Malkoc. Conn reported to the Devil's farm team in Albany, New York where he erupted for a 39 goal season, but was never given a shot with the Devils in their Stanley Cup year. Thanks in large part to Rob, the Devils farm team won the Calder Cup as the AHL champions as well.
The Sabres were quick to pluck Conn from the Devils in 1995 Waiver Draft acquisition, and he started the year in Buffalo. In fact, less than 2 weeks after being picked up by the Sabres, Rob scored his first NHL goal - and it came against his former team New Jersey!
After starting the season with the Sabres, Rob was demoted to Rochester of the AHL after his NHL playing time was greatly reduced and he was often a healthy scratch. He continued to star in the AHL and earned a return to the NHL later in the season, although he was ultimately returned to the AHL for good late in the year.
That 1995-96 season was a great season for Rob. In addition to his 28 games in the NHL where he registered 2 goals and 5 assists, he put in a great effort in the AHL and led the Amerks to the Calder Cup championship. Rob is one of only a few players to win back to back AHL championships with two different teams.
Rob, who is a cousin of long time NHLer Dave Barr, became a free agent in the summer of 1996 and opted to return to the Chicago organization. He spent the entire year back in Indianapolis before retiring from professional hockey.