The company said it had taken the decision "to structure the company to an appropriate size that is sustainable in a challenging market place".
It said it would complete the steelwork for Wembley and other landmark structures, such as the Carquinez toll bridge in California. "Cleveland Bridge has just reached the halfway mark for the erection of the deck steelwork on the world's longest cable stay bridge, the Rion-Antirion bridge in Greece and continues to work on other projects."
Brian Rogan, deputy managing director of Cleveland Bridge, added that the company was disappointed that it had to make the redundancies.
He said: "We've been successful in the recent past but must now structure our business so that we can continue to contribute as a significant member of the Darlington community."
Rogan said the market had changed, that big projects were coming to a close and that there were too few such projects available in the short term.