The MLBPA's executive board said the 41-year-old former Detroit Tigers slugger got its unanimous approval just 11 days after Weiner lost a 15-month battle against an inoperable brain tumor.
"It is with a very heavy heart that I thank the players for providing me the honor and opportunity to follow my mentor and friend, Michael Weiner, as the MLBPA's next executive director," Clark said in a statement.
"We all wish this decision was not necessary; however, we also know that Michael would urge all of us involved with the union to forge ahead in our efforts to protect the rights of the players."
"Tony has some big shoes to fill, but we're more than confident that he has the knowledge, experience and passion to serve as our next executive director," MLBPA association representative and New York Yankees outfielder Curtis Granderson added.
Clark joined the MLBPA staff in 2010 as director of player relations and was promoted in July to deputy executive director.
A second-overall pick by the Tigers in the 1990 MLB rookie draft, he played 15 seasons for Detroit, Arizona, the New York Mets and Yankees, Boston and San Diego, finishing with 1,188 hits, 251 home runs and 824 RBI over 1,559 games.