Paul who was running the coffee chain in the three years up to its £3.9bn takeover by Pepsi Cola will take over from the start of April and join the board in May.
His appointment marks a complete change at the top for the takeaway group following the appointment of Matt Shattock as chairman earlier in the month.
Domino’s is still involved in a long-running dispute with its major UK franchisees over how profits are divided and one of Paul’s first tasks will be to try to resolve that.
Last week, broker Liberum suggested that the coronavirus lockdown might "put the relationship between the Plc and franchisees at breaking point".
Domino’s is also trying to extract itself from Norway, Switzerland and Iceland, where sales have slumped and it is running up heavy losses.
Paul said: "I am hugely excited to be joining Domino's.
"It is a much-loved brand with great people and some of the best franchisees in the world, and there is a strong foundation to build on.”