Osborne was Envoy Extraordinary and Minister Plenipotentiary to the Holy See 1936–1947. His appointment came on the heels of Cardinal Secretary of State Pacelli's complaints of the short tenure of British Ministers to the Vatican; in fact, Osborne himself waited six months after his appointment before arriving in Rome. When Italy declared war on the United Kingdom in 1940, Osborne, accredited to the Holy See but living in Italian territory, moved inside the Vatican according to arrangements made under the Lateran Treaty. With a few exceptions, Osborne would be immured inside the Vatican until the liberation of Rome in 1944, working under difficult conditions. Using the code name "Mount", he was one of the group, which he supported with his own money, led by Monsignor Hugh O'Flaherty and a French diplomat François de Vial who helped conceal some 4,000 escapees, whether Jews or Allied soldiers, from the Nazis: 3,925 survived the war. Their story was portrayed in the 1983 filmThe Scarlet and the Black, starring Gregory Peck. He also played a key part in a plot in 1940, which involved the Pope and certain German generals, to overthrow Hitler. Major Sam Derry, in his book The Rome Escape Line, described meeting Sir D'Arcy in the Vatican in 1943:
Unruffled poise... Seldom have I met any man in whom I had such immediate confidence. He welcomed us warmly, yet I found it impossible to behave with anything but strict formality. Apart from the restraining influence of my clothing I was almost overwhelmed by an atmosphere of old-world English courtliness and grace which I had thought belonged only to the country-house parties of long ago. Sir D'Arcy was spry, trim, a young sixty, but he had spent years enough in the diplomatic service to develop an astonishing aptitude for creating around himself an aura of all that was most civilized in English life. I felt as though I had returned home after long travels, to find that royalty had come to dinner, and I had to be on my best behaviour.
Following this dinner Sir D'Arcy offered Derry the command of the escape organisation.
Post-war
After the war Osborne retired from the Diplomatic Service and settled in Italy, living at the Palazzo Sacchetti, 66 Via Giulia, Rome. He succeeded his second cousin once removed as Duke of Leeds in 1963. He died just a year later, at the age of seventy-nine, at which point the dukedom and all of its subsidiary titles became extinct. The Duke of Leeds is buried in the Protestant Cemetery, Rome.
Osborne kept an extensive diary, portions of which were used by Owen Chadwick as the basis of his 1980 Ford Lectures and his 1988 book, Britain and the Vatican during the Second World War. Chadwick's quotations from Osborne's diary included: "I reached the grave conclusion during the Mass that I am nothing but a pencilled marginal note in the Book of Life. I am not in the main text at all."