Cubitt’s move involves a swap with Scot Marchbank, who took on the deputy spokesman role on a permanent basis last month.
It comes as the Conservatives position themselves as tough on criminals, with Secretary of State Chris Grayling last weekend scrapping the use of cautions for offences such as rape, manslaughter and robbery.
He previously worked for Grayling as a press secretary when Grayling was minister for employment, and during a five-year stint for Cubitt at the Department for Work & Pensions (DWP) between 2006 and 2011.
Cubitt, hired after a recruitment process, will report to director of comms Pam Teare when he takes on the role in a few weeks' time.
He pointed to policy developments at the department, including rehabilitation of offenders and legal aid reform, as key drivers behind his interest in the job.
"They’re leading some cutting-edge public service reform," Cubitt said.
"That was something I enjoyed doing at the DWP, and you’ve also got a great team and secretary of state there."
Cubitt left the Cabinet Office to take on the interim deputy spokesman role after the departure of Vickie Sheriff earlier this year.