The 53-year-old moved to the Commerzbank Arena in 2011 and led the team back to the top flight in his first season in charge.
Frankfurt followed that achievement by placing sixth in last season's Bundesliga table, a return that earned a subsequent spot in the UEFA Europa League.
However, Frankfurt have struggled to replicate that form this term and currently sit 12th, just six points above the relegation zone, while the club were knocked out of Europe last week by Porto.
Veh revealed on Monday that he intended to make the announcement sooner, but respected the club's decision to hold fire.
"I wanted to make the decision public in the winter but accepted the club's wish to do so at a later date," he said.
"It's very important to me that, in spite of all the problems we may have, we achieve our aim of staying in the Bundesliga.
"It was also important for me to inform the club well in advance, which I have done. I only told the players of my decision today.
"I cannot imagine that my departure has a negative impact on the team."
Frankfurt president Heribert Bruchhagen added: "Armin Veh informed us in Abu Dhabi (where the club had a training camp during the mid-season break) that he will not be extending his contract here.
"We are all determined to continue working successfully with Armin Veh until the end of June.
"There have been no discussions with other coaches."
Frankfurt may have had struggles in Germany's top flight this season, but they did secure a 2-1 victory over Stuttgart on Sunday thanks to late goals from Jan Rosenthal and Alexander Meier.
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