Cossa, who has run the Cooperstown, New York, company since April 2003, is struggling with the illnesses of two family members, she said in a statement.
"With regret and sadness, I have concluded that personal circumstances...prevent me from giving my full attention to the demanding job of general director," she said. "I shall miss terribly devoting myself to a company I have always admired and have come to love."
Managing director Jeryl Dropp and director of finance Paul Griswold will oversee the company while a search committee seeks a replacement for Cossa, board chairman Robert Schlather said.
"The Board of Trustees has accepted Joanne's resignation with great regret," Schlather added. "She has led the company during two of the most successful seasons in its history."
The former executive vice-president of Symphony Space in New York City, Cossa succeeded Esther Nelson, who retired at the end of the 2002 season.
Glimmerglass Opera stages several operas each summer at the Alice Busch Opera Theater, on the shores of Otsego Lake‹the "Glimmerglass" of James Fenimore Cooper's Leatherstocking Tales. The highly regarded company shares artistic director Paul Kellogg with New York City Opera, and often collaborates on new productions with NYCO.