The Pole will drive for Williams Racing in the upcoming 2019 season, joining the British team alongside George Russell.
"Firstly, I would like to thank everyone who has helped me during what was a difficult period of my life over these last few years," Kubica said.
"It has been a challenging journey to make it back to the Formula One grid, but what seemed almost impossible is now beginning to feel possible.
"For sure, it has been a long road to get to this point, but as that challenge now comes to an end with this announcement, a new challenge begins working with Williams on track."
Kubica was competing in the Ronde di Andora rally championship in 2011 when he lost control of his car at high speed and smashed into a roadside barrier.
The then 26-year-old, considered one of the most exciting prospects in the sport, was trapped for more than an hour before rescue workers managed to free him, allowing him to be airlifted to hospital.
It was later confirmed he had required a partial amputation of his right forearm, while also suffering severe injuries to his right elbow, shoulder and leg.