He will take a new job as director of comms at the North West London Collaborative of Clinical Commissioning Groups – a group of eight CCGs in the capital.
Cubitt has been deputy spokesman to Prime Minister David Cameron since September, joining from an interim role in the Cabinet Office.
He had previously worked in Downing Street during the previous coalition government, and at the Ministry of Justice.
Cubitt said: "As we move to deliver a truly exceptional 24 hours a day, seven days a week service, comms will play a key role. I am really looking forward to taking on this challenge and working with the team in North West London to make sure we are the gold standard."
Cameron's spokeswoman Helen Bower and Alex Aiken, executive director for UK government comms, are recruiting for a new deputy spokesperson.
According to the job advertisement on PRWeek – which closed to further applications last week – the role is a two-year fixed-term appointment with the possibility of extension.