The statement said Qatar had provided "direct and indirect assistance to terrorist organisations in Somalia and other areas".
But it also noted that Qatar had consistently rejected claims that it was supporting anti-Ethiopian groups.
The statement said that "all diplomatic efforts have been deployed to try and persuade Qatar to change its destructive activities," but that Ethiopia now had "no alternative" but to cut ties with Qatar.
"Apparently blinded by arrogance, the source of which is clear enough, Qatar has remained deaf to all our efforts," the statement said.
There has been no official response from Qatar.
Media worries
Ethiopia also said that its complaints included its support for Eritrea and "the output of its media outlets".
In recent days Al Jazeera has broadcast several reports on the conflict in Ethiopia's Ogaden region, where ethnic Somali rebels are fighting government forces.

"Al Jazeera is using inaccurate and misleading information, fabricated by opposition elements backed by a state which makes no secret of its efforts to destabilise not only Ethiopia but also the entire sub region," the Ethiopian foreign ministry alleged in a statement.
The Ethiopian authorities have imposed a news blackout on the region, which is populated by Somali-ethnic Muslims and has restricted humanitarian work in the area.