The Minister for Education will her seat at the next election she changes her stance on according to the president of na Gaeilge, Mr Gearoid O Caireallain.
He told the organisation's, ardfheis in Belfast on Saturday he would stand as a "spoiler" candidate against Ms Breathnach in Dun Laoghaire unless three key demands were met.
The Conradh was seeking a reversal of changes in the rules governing primary teachers, which allowed teachers from the North, and elsewhere to take up posts without a qualification in Irish.
It also sought a separate education board for all Irish and Gaeltacht schools and a commitment to end "the longstanding campaign against Irish" within the Department, he said.
In a statement aimed directly at Ms Breathnach, Mr O Caireallain said: "This is not a lot of big talk to the wind but a simple threat I will stand against you, Minister, at the next general election and you know, I won't win the seat will do enough to ensure that you don't win it either.
He said language activists were "sick and tired" of oppositions within the Department to the growth of all Irish schools. "The Department doesn't understand, or chooses not to understand, the importance of a hoard as a support for Irish medium education. Step by step, year after year, the Department is chipping away at Irish."
About 150 delegates attended the ardfheis, which followed a year of significant change within the Conradh. Although the organisation has languished in the doldrums for most of the past two "decades, it has grown rapidly over the past year, especially in the North.
The growth is concentrated in Belfast, where 18 new branches have been set up. The greatest change, however, has been the lowering of the age profile in the executive with the election, at the ardfheis, of a number of younger activists. Mr O Caireallain was returned unopposed as president.