Production at the Redcar plant was paused on 18 September with the company citing poor steel trading conditions and a drop in world steel prices.
SSI UK business director Cornelius Louwrens said: "This is an extremely sad day for all of us at SSI UK."
Redcar MP Anna Turley described it as "devastating news".
SSI said it had carried out an assessment of the business situation it faced and had concluded there was "no other option" but to mothball its iron and steel making facilities on Teesside.
The Redcar coke ovens and the power station will continue to operate.