The vessel will be powered by liquefied natural gas (LNG) and will come with other environmentally friendly features, according to Royal.
Two sister Icon-class ships will be delivered in 2022 and 2024, each with capacity for around 5,000 passengers.
Finnish shipbuilder Meyer Turku will build the vessel.
The line says its Icon-class ships are a “breakthrough” in ship design, featuring various “clean technologies”.
Royal chairman and chief executive Richard Fain said the order “affirmed Royal’s commitment to clean-power technologies at sea”.
"We’ve designed a class of ships powered by liquefied natural gas that leverages the latest, environmentally friendly applications,” said Fain.
“We believe innovative shipbuilding can reduce our carbon footprint and boost energy efficiencies to help to build a cleaner future."
President and chief executive Michael Bailey added: "Building on our vision with the addition of a third order is a testament to our confidence in the innovative design and energy-efficient technology and engineering that will, without a doubt, make Icon class a game changer."
Royal has stressed the order is contingent on financing, although at this stage the line has revealed no details of the value of its Icon-class order.
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