Prosecutors say that more than 400 deaths were caused by the toxic emissions from the Rivas' Ilva steel firm.
Heavily indebted and short of cash, Ilva was taken over by the Italian state in 2014.
The year before it had been placed under special administration after the Rivas were accused of failing to prevent toxic emissions including carcinogenic particles from spewing out across the town of Taranto in the Puglia region of southern Italy.
Prosecutors say that more than 400 deaths were caused by the toxic emissions.
The family admitted to having saved over 1.3 billion euros between 2009 and 2013 by delaying work to bring the steel plant into compliance with safety and environmental norms, and agreed to restitute the funds to help clean up the facility and keep it running.
Adriano Riva, whose brother Emilio ran the company until his death in 2014, on Wednesday signed the documents to transfer to Italy 1.3 billion euros from Switzerland, where the funds had been frozen as part of the investigation into the company.
Swiss authorities are expected to clear the transfer next week.
After the transfer papers were signed a judge validated a deal with prosecutors for Riva to serve two and a half years in prison.
Cases against two of Emilio's sons, Nicola and Fabio, are still at a preliminary stage.
The Italian state launched last year an appeal for offers for Ilva. Two binding offers were submitted in March, one by global steel giant ArcelorMittal which teamed up with Italian firm Marcegaglia, and the other by India's JSW Steel with a group of Italian companies.